I think expanding means to remove the parentheses/brackets from a problem.
For example: Say we have the expression: 3 (4 + 5). I think expanding means to multiply 3, by every number in the parentheses. So that means:
(3 * 4) + (3 * 5) = 27.
Another way to think about it is to (if you're on paper) draw a line from 3, to all the numbers inside the parentheses. The line that connects from 3 to 4, is signaling for you to multiply 3 * 4 = 12. And the line from 3 to 5 = 3 * 5 = 15. And add them.
Final answer:
In mathematics, 'expand' refers to writing an expression in an extended form using distribution. This can result in a polynomial or an infinite series, as seen in binomial expansion or exponential arithmetic.
Explanation:
In mathematics, to expand means to increase the length of an expression by distributing multiplication over addition or subtraction. For example, expanding (a + b)(c + d) results in ac + ad + bc + bd. This does not change the value of the expression, but rather writes it in an alternative form that might be more useful for further operations, such as simplification or evaluation. Binomial expansion, specifically, refers to expressing a binomial raised to a power as a series of terms, using the binomial theorem, which can sometimes result in an infinite series or a polynomial of finite length. This expansion is applicable in situations like expanding (x + y)^n or when dealing with power series expansions of standard mathematical functions including exponential arithmetic where numbers are expressed as a product of a digit term and an exponential term such as in the notation 4.57 x 10^3.
Determine whether the given description corresponds to an experiment or an observational study. A stock analyst selects a stock from a group of twenty for investment by choosing the stock with the greatest earnings per share reported for the last quarter.A) Experiment B) Observational study
The description corresponds to an observational study as the analyst is merely observing and analyzing the existing data (earnings per share) to make an investment decision, there's no control or manipulation of the variables involved.
Explanation:The given description corresponds to an observational study. This is because the stock analyst is merely observing and analyzing the existing earnings per share of the stocks from a group of twenty and then making an investment decision based on this data. There is no manipulation or control of variables, which are defining characteristics of an experiment.
In an experiment, the researchers would have actively influenced the earnings per share (the variable) in some way to gauge the effect of that influence. However, in this case, the analyst is simply observing the earnings per share as they are to select a stock for investment.
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The deck for a card game is made up of 108 cards. Twenty-five each are red, yellow, blue, and green, and eight are wild cards. Each player is randomly dealt a seven-card hand.
(a) What is the probability that a hand will contain exactly two wild cards?
(b) What is the probability that a hand will contain two wild cards, two red cards, and three blue cards?
(a) The probability that a hand will contain exactly two wild cards is 0.076.
(b) The probability that a hand will contain two wild cards, two red cards, and three blue cards is 0.0007.
Let's solve these problems using the concept of combinations in probability.
Remember, [tex]C(n, k)[/tex] denotes the number of ways to choose k items from a set of n items, and is calculated as
[tex]C(n,k)=\frac{n!}{k!(n-k)!}[/tex]
where "!" denotes factorial. For example, [tex]5! = 5 \times 4 \times 3 \times 2 \times 1[/tex]
(a) The probability that a hand will contain exactly two wild cards:
The total number of ways to choose 7 cards out of 108 is [tex]C(108, 7)[/tex].
The number of ways to choose 2 wild cards out of 8 is [tex]C(8, 2)[/tex].
The number of ways to choose the remaining 5 cards out of the 100 non-wild cards is [tex]C(100, 5)[/tex].
So, the probability is [tex]\frac{C(8, 2) \times C(100, 5)}{C(108, 7)} \approx 0.076[/tex]
(b) The probability that a hand will contain two wild cards, two red cards, and three blue cards:
The number of ways to choose 2 wild cards out of 8 is [tex]C(8, 2)[/tex].
The number of ways to choose 2 red cards out of 25 is [tex]C(25, 2)[/tex].
The number of ways to choose 3 blue cards out of 25 is [tex]C(25, 3)[/tex].
So, the probability is [tex]\frac{C(8, 2) \times C(25, 2) \times C(25, 3)}{C(108, 7)} \approx 0.0007[/tex]
Two solids are described in the list below.
One solid is a sphere and has a radius of 6 inches.
The other solid is a cylinder with a radius of 6 inches and a height of 6 inches.
what is the difference betwen the volumes in cubic inches of the solids in terms of pi
A.72pi
B.144pi
C.216pi
D.288pi
The difference between the volumes in cubic inches is option A) 72pi
Step-by-step explanation:
Volume of the sphere = 4/3 πr³radius r = 6 inchesVolume = 4/3 π(6)³
⇒ 4/3(216)π
⇒ 4[tex]\times[/tex]72π
⇒ 288π cubic inches
Volume of a cylinder = π r²hradius r = 6 inchesheight h = 6 inchesVolume = π(6)²(6)
⇒ 6³π
⇒ 216π cubic inches
Difference between the volumes = 288π - 216π = 72π
The difference in volume between the two solids is 226.08in^3
Data;
radius of sphere = 6inradius of cylinder = 6inheight of cylinder = 6inVolume of SphereThe volume of a sphere is given as
[tex]v = \frac{4}{3} \pi r^3\\[/tex]
Let's substitute the values and find the volume
[tex]v = \frac{4}{3}*3.14*6^3\\v = 904.32in^3[/tex]
Volume of CylinderThe formula of volume of a cylinder is given as
[tex]v = \pi r^2 h\\[/tex]
Let's substitute the values into the equation and solve
[tex]v = 3.14 * 6^2 * 6\\v = 678.24in^3[/tex]
The difference in volume between the two solids is
[tex]volume of sphere - volume of cylinder = 904.32 - 678.24 = 226.08in^3[/tex]
The difference in volume between the two solids is 226.08in^3
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Which coordinate plane shows the graph of 3x + y > 9?
Answer:
'3'. x = 3 + 0.3333333333y Simplifying x = 3 + 0.3333333333y
Step-by-step explanation:
Suppose an individual makes an initial investment of $3,000 in an account that earns 6.6%, compounded monthly, and makes additional contributions of $100 at the end of each month for a period of 12 years. After these 12 years, this individual wants to make withdrawals at the end of each month for the next 5 years (so that the account balance will be reduced to $0). (Round your answers to the nearest cent.)
(a) How much is in the account after the last deposit is made?
(b) How much was deposited?
(c) What is the amount of each withdrawal?
(d) What is the total amount withdrawn?
I get A and C. If you could explain B and D I'd appreciate it.
Answer:
b) $17,400
d) $33,517.20
Step-by-step explanation:
a) $28,482.19 . . . . future value of all deposits
__
b) The initial deposit was $3000, and there were 144 deposits of $100 each, for a total of ...
$3000 +144×100 = $17,400 . . . . total deposited
__
c) $558.62
__
d) 60 monthly withdrawals were made in the amount $558.62, for a total of ...
60×$558.62 = $33,517.20 . . . . total withdrawn
_____
Additional information about (a) and (c)
(a) The future value of the initial deposit is the deposit multiplied by the interest multiplier over the period.
A = P(1 +r/n)^(nt) = 3000(1 +.066/12)^(12·12) = 3000·1.0055^144 ≈ 6609.065
The future value of $100 deposits each month is the sum of the series of 144 terms with common ratio 1.0055 and initial value 100.
A = 100(1.055^144 -1)/0.0055 ≈ 21,873.123
So, the total future value is ...
$6609.065 +21873.123 ≈ $28482.188 ≈ $28,482.19
__
(c) The withdrawal amount can be found using the same formula used for loan payments:
A = P(r/n)/(1 -(1 +r/n)^(-nt)) = $28482.19(.0055)/(1 -1.0055^-60) ≈ $558.62
The total amount deposited in the account was $17,400 including an initial investment of $3,000 and subsequent monthly payments of $100 for 12 years. The total amount withdrawn was equal to the final balance after the last deposit.
Explanation:Let's tackle each question one by one:
You've mentioned that you have already figured out part (a) and (c), so let's move on to part (b).(b) How much was deposited?The individual started with an initial deposit of $3,000. After that, they deposited $100 at the end of each month for 12 years. That's 12 years * 12 months/year * $100/month, for a total of $14,400. So, if you add the initial deposit, the total amount deposited over the whole period is $3,000 + $14,400 = $17,400.(d) What is the total amount withdrawn?The total amount withdrawn is the same as the final balance of the account after the last deposit, as the question states the account balance will be zero after the withdrawals. Since you have already figured out part (a) which is the account balance after the last deposit, the total amount withdrawn corresponds to that sum.Learn more about Compound Interest here:
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Consider a railroad bridge over a highway. A train passing over the bridge dislodges a loose bolt from the bridge, which proceeds to fall straight down and ends up breaking the windshield of a car passing under the bridge. The car was 25 m away from the point of impact when the bolt began to fall down; unfortunately, the driver did not notice it and proceeded at constant speed of 21 m/s. How high is the bridge
Answer:
the bridge has a height y₀ = 6.94 m
Step-by-step explanation:
The position y of the loose bolt is given by (0,y) where
y = y₀ - 1/2*g*t²
where
y₀ = initial position of the bolt (height of the bridge) , g= gravity , t=time
and the position x of the car is given by (x,0) where
x= x₀ + v*t
where
x₀= initial position of the car
v= car's velocity
then in order for the bolt to hit the windshield they should be at x=0 and y=0 at the same time , then
0= x₀ + v*t
t= -x₀/v
replacing in the equation for y
0 = y₀ - 1/2*g*t²
0 = y₀ - 1/2*g*(-x₀/v)²
0 = y₀ - 1/2*g*x₀²/v²
y₀ = 1/2*g*x₀²/v²
replacing values
y₀ = 1/2*g*x₀²/v² = 1/2* 9.8m/s² * (-25 m)²/(21 m/s)² = 6.94 m
then the bridge has a height y₀ =6.94 m
We have assumed that
- The bolt has no horizontal velocity ( only vertical velocity) , starts from rest and neglected air friction
- Neglecting the height of the car , position of the windshield and size of the loose bolt
Estimate the instantaneous rate of change of h (t) = 2t² + 2 at the point t = −1.
In other words, choose x-values that are getting closer and closer to −1 and compute the slope of the secant lines at each value. Then, use the trend/pattern you see to estimate the slope of the tangent line.
Answer:
The instantaneous rate of change of h(t) = 2t² + 2 at the point t = −1 is -4.
Step-by-step explanation:
If two distinct points [tex]P(x_1,y_1)[/tex] and [tex]Q(x_2,y_2)[/tex] lie on the curve [tex]y=f(x)[/tex], the slope of the secant line connecting the two points is
[tex]m_{sec}=\frac{y_2-y_1}{x_2-x_1}=\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]
If we let the point [tex]x_2[/tex] approach [tex]x_1[/tex], then Q will approach P along the graph f(x). The slope of the secant line through points P and Q will gradually approach the slope of the tangent line through P as
[tex]m_{tan}= \lim_{x_2 \to x_1}\frac{f(x_2)-f(x_1)}{x_2-x_1}[/tex]
And this is the instantaneous rate of change of the function f(x) at the point [tex]x_1[/tex].
From the information given, we know that the point P [tex](-1,2(-1)^2+2)=(-1,4)[/tex] lies on the curve [tex]h(t) = 2t^2 + 2[/tex].
If Q is the point [tex](t, 2t^2 + 2)[/tex] we can find the slope of the secant line PQ for the following values of t. Because we choose values that are getting closer and closer to −1.
[tex]\begin{array}{c}-0.9&-0.99&-0.999&-0.9999\\-1.1&-1.01&-1.001&-1.0001\\\end{array}\right[/tex]
Let the point P be [tex](x_2=-1, y_2=4)[/tex] and the point Q be [tex](x_1=t, y_1=2t^2+2)[/tex]. So,
[tex]m=\frac{4-(2t^2+2)}{-1-t}\\\\m=-\frac{2\left(t+1\right)\left(t-1\right)}{-1-t}\\\\m=2\left(t-1\right)[/tex]
Next, substitute the value of x in the formula of the slope
[tex]m=2(-0.9-1)=-3.8[/tex]
Do this for the other values of x.
Below, there is a table that shows the values of the slope.
From the table, as t approaches -1 from the left side (-0.9 to -0.9999), the slopes are approaching to -4 and as t approaches -1 from the right side (-1.1 to -1.0001), the slopes are approaching to -4. The value of the slope at P(-1,4) is then m = -4.
Final answer:
To estimate the instantaneous rate of change of h(t) at t = -1, we calculate the slopes of secant lines near that point and observe the pattern to approximate the slope of the tangent line, which represents the instantaneous velocity.
Explanation:
To estimate the instantaneous rate of change of the function h(t) = 2t² + 2 at t = −1, we need to calculate the slope of the tangent line at that point. We can approximate this slope using secant lines connecting points increasingly closer to t = −1.
Let's select two points close to t = −1, say t1 = −1.1 and t2 = −0.9, and compute the slope of the secant line:
Slope of secant line = (h(t2) − h(t1)) / (t2 − t1) = (3.62 − 4.42) / (-0.9 + 1.1) = −0.8 / 0.2 = −4.
To get a more accurate approximation, we'd choose points even closer to t = −1 and observe the pattern. We can assume that the slope of the tangent line, which represents the instantaneous velocity, is approximately equal to the slopes of the secant lines as they converge to a single value.
The article "Chances are you know someone with a tattoo, and he's not a sailor" included results from a survey of adults aged 18 to 50. The accompanying data are consistent with the summary values given in the article. Assuming these data are representative of adult Americans and that an adult American is selected at random, use the given information to estimate the following probabilities.
(A) P(tattoo)
(B) P(tattoo | age 18-29)
(C) P(tattoo | age 30-50)
(D) P(age 18-29 | tattoo)
At Least One Tattoo No Tattoo
Age 18-29 126 324
Age 30-50 54 396
Answer:
a) 0.2
b) 0.28
c) 0.12
d) 0.7
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
We have the following table:
At Least One Tattoo No Tattoo
Age 18-29 126 324
Age 30-50 54 396
So in total, there are
126 + 324 + 54 + 396 = 900 people
(A) P(tattoo)
This is the probability that a randomly selected person has a tattoo.
Desired outcomes:
126 + 54 = 180
180 people have at least one tattoo
Total outcomes:
There are 900 people.
P(tattoo) = 180/900 = 0.2
(B) P(tattoo | age 18-29)
This the probability that a person aged 18-29 has a tattoo
Desired outcomes:
126 people aged 18-29 have tattoos
Total outcomes:
126 + 324 = 450 people aged 18-29
P(tattoo | age 18-29) = 126/450 = 0.28
(C) P(tattoo | age 30-50)
This the probability that a person aged 30-50 has a tattoo
Desired outcomes:
54 people aged 18-29 have tattoos
Total outcomes:
54 + 396 = 450 people aged 30-50
P(tattoo | age 18-29) = 54/450 = 0.12
(D) P(age 18-29 | tattoo)
The probability that a tattoed person is 18-29.
Desired outcomes:
126 tattoed people are 18-29
Total outcomes
126 + 54 = 180 tattoed people
P(age 18-29 | tattoo) = 126/180 = 0.7
An urn contains 13 red balls and 7 blue balls. Suppose that three balls are taken from the urn, one at a time and without replacement. What is the probability that at least one of the three taken balls is blue?
Answer:
0.749
Step-by-step explanation:
The probability that at least one of the three taken balls is blue is the inverse of the probability that none of the three taken balls is blue, aka all 3 of the taken balls are red. The probability of this to happen is
In the first pick: 13/20 chance of this happens
In the 2nd pick: 12/19 chance of this happens
In the 3rd pick: 11/18 chance of this happens
So the probability of picking up all 3 red balls is
[tex]\frac{13*12*11}{20*19*18} = \frac{1716}{6840} = 0.251[/tex]
So the probability of picking up at least 1 blue ball is
1 - 0.251 = 0.749
Keeping water supplies clean requires regular measurement of levels of pollutants. The measurements are indirect—a typical analysis involves forming a dye by a chemical reaction with the dissolved pollutant, then passing light through the solution and measuring its "absorbence." To calibrate such measurements, the laboratory measures known standard solutions and uses regression to relate absorbence and pollutant concentration. This is usually done every day. Here is one series of data on the absorbence for different levels of nitrates. Nitrates are measured in milligrams per liter of water.
Nitrates 50 50 100 200 400 800 1200 1600 2000 2000
Absorbence 7.0 7.6 12.7 24.0 47.0 93.0 138.0 183.0 231.0 226.0
The calibration process sets nitrate level and measures absorbence. The linear relationship that results is used to estimate the nitrate level in water from a measurement of absorbence.
a. What is the equation of the line used to estimate nitrate level?
b. What does the slope of this line say about the relationship between nitrate level and absorbence?
c. What is the estimated nitrate level in a water specimen with absorbence 40?
Answer:
a) Equation is
[tex]y = 0.1135x+1.590[/tex]
b) Slope = 0.1135 represents the change in y for a unit change in x
i.e. When nitrate content is increasedby 1, absorbence is increased by 0.1135
Step-by-step explanation:
Nitrates Absorbence
x y
50 7
50 7.6
100 12.7
200 24
400 47
800 93
1200 138
1600 183
2000 231
2000 226
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.999911043
R Square 0.999822094
Adjusted R Square 0.999799856
Standard Error 1.2890282
Observations 10
Coefficients
Intercept 1.589782721
x 0.113500259
we get regression line as
y = 0.1135x+1.590
a) Equation is
[tex]y = 0.1135x+1.590[/tex]
b) Slope = 0.1135 represents the change in y for a unit change in x
i.e. When nitrate content is increasedby 1, absorbence is increased by 0.1135
The U.S. Bureau of Labor Statistics reports that 11.3% of U.S. workers belong to unions (BLS website, January 2014). Suppose a sample of 400 U.S. workers is collected in 2014 to determine whether union efforts to organize have increased union membership. a. Formulate the hypotheses that can be used to determine whether union membership increased in 2014. H0: p Ha: p b. If the sample results show that 52 of the workers belonged to unions, what is the p-value for your hypothesis test (to 4 decimals)? c. At α = .05, what is your conclusion?
Answer:
a) Null hypothesis: [tex] p \leq 0.113[/tex]
Alternative hypothesis: [tex] p >0.113[/tex]
b) [tex]p_v =P(z>1.07)=0.1423[/tex]
c) So the p value obtained was a high low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL reject the null hypothesis, and we can said that at 5% of significance the proportion of workers belonged to unions is not significantly higher than 0.113.
Step-by-step explanation:
Part a
For this case we want to check the following system of hypothesis:
Null hypothesis: [tex] p \leq 0.113[/tex]
Alternative hypothesis: [tex] p >0.113[/tex]
Part b
Data given and notation
n=400 represent the random sample taken
X=52 represent the workers belonged to unions
[tex]\hat p=\frac{52}{400}=0.13[/tex] estimated proportion of workers belonged to unions
[tex]p_o=0.113[/tex] is the value that we want to test
[tex]\alpha=0.05[/tex] represent the significance level
Confidence=95% or 0.95
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
Concepts and formulas to use
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.13 -0.113}{\sqrt{\frac{0.113(1-0.113)}{400}}}=1.07[/tex]
Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
The significance level provided [tex]\alpha=0.05[/tex]. The next step would be calculate the p value for this test.
Since is a right tailed test the p value would be:
[tex]p_v =P(z>1.07)=0.1423[/tex]
Part c
So the p value obtained was a high low value and using the significance level given [tex]\alpha=0.05[/tex] we have [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL reject the null hypothesis, and we can said that at 5% of significance the proportion of workers belonged to unions is not significantly higher than 0.113.
The time taken for a computer to boot up, X, follows a normal distribution with mean 30 seconds and standard deviation 5 seconds. What is the probability that a computer will take more than 42 seconds to boot up?
Answer:
0.008 is the probability that a computer will take more than 42 seconds to boot up.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 30 seconds
Standard Deviation, σ = 5 second
We are given that the distribution of time taken for a computer to boot up is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) P(computer will take more than 42 seconds to boot up)
P(x > 42)
[tex]P( x > 42) = P( z > \displaystyle\frac{42 - 30}{5}) = P(z > 2.4)[/tex]
[tex]= 1 - P(z \leq 2.4)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x > 42) = 1 - 0.992 = 0.008[/tex]
0.008 is the probability that a computer will take more than 42 seconds to boot up.
The tip of a fisherman’s rod is 8 feet above the surface of the water when he catches a fish. If he reels in a fish at a rate of 1 foot per second, and never moves the position of the rod, at what rate is the fish approaching the base of the dock when 10 feet of fishing line is out?
Answer:
-1.28 ft/s
Step-by-step explanation:
We are given that
The height of tip of fisherman's rod from the water surface=y=8 ft
[tex]\frac{dz}{dt}=-1ft/sec[/tex]
We have to find the rate at which the fish is approaching the base of the dock when x=10 ft
[tex]z=\sqrt{x^2+y^2}[/tex]
By Pythagoras theorem
[tex]Hypotenuse=\sqrt{base^2+(perpendicular\;side)^2}[/tex]
Substitute x=10 and y=8
[tex]z=\sqrt{(10)^2+8^2}=\sqrt{164}=2\sqrt{41}ft[/tex]
[tex]x^2+y^2=z^2[/tex]
Differentiate w.r.t t
[tex]2x\frac{dx}{dt}+2y\frac{dy}{dt}=2z\frac{dz}{dt}[/tex]
[tex]x\frac{dx}{dt}+y\frac{dy}{dt}=z\frac{dz}{dt}[/tex]
Substitute the values
[tex]10\frac{dx}{dt}+8(0)=2\sqrt{41}\times (-1)[/tex]
[tex]\frac{dy}{dt}=0[/tex]
Because he never moves the rod.
[tex]\frac{dx}{dt}=\frac{-2\sqrt{41}}{10}=-1.28 ft/s[/tex]
Hence, the fish is approaching the base of the dock at the rate of 1.28 ft/s
The information for 2008 in millions in the table below was reported by the World Bank. On the basis of this information, which list below contains the correct ordering of real GDP per person from highest to lowest? Country GDP (Constant USS) GDP(Current USS) Population Germany 2,091 573 3,649,493 82.11 Japan 5,166,281 4910,839 127.70 U.S 11,513,872 14,093.309 304.06 A. Japan, Germany, United States B. Japan, United States, Germany C. Germany, United States, Japan D. Unied States, Japan. Germany
Answer:
Option D
Step-by-step explanation:
The current GDP is a true reflective of the actual GDP per person.
The average GDP per person is given as follows:
average GDP = [tex]\frac{Current GDP}{total population}[/tex]
For example, take Germany:
Amount in millions ( current GDP) = 3,649,493
Total population = 82 110 000
GDP per person = [tex]\frac{3649493}{82110000}[/tex]
= 0.044
The list in the descending order will be:
U.S
Japan
Germany
The correct ordering of real GDP per person is Japan, Germany, United States.
Explanation:The correct ordering of real GDP per person from highest to lowest based on the given information is Japan, Germany, United States (option B).
GDP per person is calculated by dividing the GDP (Constant USS) by the population. In this case, for 2008, the GDP per person for Japan is 5,166,281 / 127.70 = 40,442.37, for Germany is 2,091,573 / 82.11 = 25,467.29, and for the United States is 11,513,872 / 304.06 = 37,868.49.
Therefore, option A) Japan has the highest real GDP per person, followed by the United States, and then Germany.
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Explain the meaning of each of the following. (a) lim x → −3 f(x) = [infinity] The values of f(x) can be made arbitrarily close to 0 by taking x sufficiently close to (but not equal to) −3. The values of f(x) can be made arbitrarily large by taking x sufficiently close to (but not equal to) −3.
Answer:
The right answer is option 3.
lim x → −3 f(x) = [infinity] means the values of f(x) can be made arbitrarily large by taking x sufficiently close to (but not equal to) −3.
Step-by-step explanation:
The limit of a function is a fundamental concept concerning the behavior of that function near a particular input.
A function f assigns an output f(x) to every input x. We say the function has a limit L at an input a: this means f(x) gets closer and closer to L as x moves closer and closer to a. More specifically, when f is applied to any input sufficiently close to a, the output value is forced arbitrarily close to L.
That is,
lim x → a f(x) = L
Hope this helps!
The limit lim x → −3 f(x) = [infinity] means that as x values get closer to -3 (without becoming -3), the value of the function f(x) goes towards infinity i.e., it grows without bound. This is akin to certain function behaviors near a value at which an asymptote is present. However, the second part about f(x) values getting close to 0 seems contradiction to the first statement.
Explanation:The statement lim x → −3 f(x) = [infinity] is related to a concept in Calculus known as a limit. When we say that the limit of f(x) as x approaches -3 is infinity, we mean that as we make x values closer and closer to -3 (without letting x actually be -3), the value of the function f(x) becomes larger and larger without bound, i.e., approaches infinity.
This is similar to some function behaviors near an asymptote. For example, the function y = 1/x has a vertical asymptote at x = 0, where y approaches infinity as x approaches zero from either direction. Here, as x gets arbitrarily close to 0, the value of y = 1/x gets arbitrarily large, or 'approaches infinity'.
On the other hand, when the question states, 'The values of f(x) can be made arbitrarily close to 0 by taking x sufficiently close to (but not equal to) -3', it signifies the tendency of the function values to get closer and closer to 0 as x gets closer to -3. This indicates a certain limit behavior, but it seems to be contradictory with the first part where the limit was stated to be infinity. It is important to scrutinize the function's properties and behavior around x = -3 carefully.
Learn more about Limit here:https://brainly.com/question/37042998
A sample of 100 cars driving on a freeway during a morning commute was drawn, and the number of occupants in each car was recorded. The results were as follows: NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Occupants 1 2 3 4 5 Number of Cars 74 10 11 3 2 Find the sample standard deviation of the number of occupants. The sample standard deviation is 37.60 37.60 Incorrect . (Round the final answer to two decimal places.)
Answer:
[tex]E(X)=1*0.74 +2*0.1 +3*0.11+ 4*0.03 +5*0.02=1.49[/tex]
[tex]Var(X)=E(X^2)-[E(X)]^2 =3.11-(1.49)^2 =0.8899[/tex]
[tex]Sd(X)=\sqrt{Var(X)}=\sqrt{0.8899}=0.943[/tex]
Step-by-step explanation:
For this case we have the following data given:
X 1 2 3 4 5
F 74 10 11 3 2
The total number of values are 100, so then we can find the empirical probability dividing the frequency by 100 and we got the followin distribution:
X 1 2 3 4 5
P(X) 0.74 0.10 0.11 0.03 0.02
Previous concepts
In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".
The variance of a random variable Var(X) is the expected value of the squared deviation from the mean of X, E(X).
And the standard deviation of a random variable X is just the square root of the variance.
Solution to the problem
In order to calculate the expected value we can use the following formula:
[tex]E(X)=\sum_{i=1}^n X_i P(X_i)[/tex]
And if we use the values obtained we got:
[tex]E(X)=1*0.74 +2*0.1 +3*0.11+ 4*0.03 +5*0.02=1.49[/tex]
In order to find the standard deviation we need to find first the second moment, given by :
[tex]E(X^2)=\sum_{i=1}^n X^2_i P(X_i)[/tex]
And using the formula we got:
[tex]E(X^2)=1^2 *0.74 +2^2 *0.1 +3^2 *0.11 +4^2 0.03 +5^2 *0.02=3.11[/tex]
Then we can find the variance with the following formula:
[tex]Var(X)=E(X^2)-[E(X)]^2 =3.11-(1.49)^2 =0.8899[/tex]
And then the standard deviation would be given by:
[tex]Sd(X)=\sqrt{Var(X)}=\sqrt{0.8899}=0.943[/tex]
Researchers estimate by the end of 2020, 850,000 new cases of heart disease and 275,000 heart disease deaths will be recorded in the United States. The estimated population of the United States at mid-point of 2020 is 341,672,244. Calculate the projected incidence rate of heart disease per 100,000 in the United States in 2020.
Answer:
249 per 100,000
Step-by-step explanation:
The projected incidence rate of heart disease per 100,000 in the United States in 2020 is determined by the number of estimated new cases of heart disease multiplied by 100,000 and divided by the estimated population of the United States in 2020:
[tex]R=\frac{850,000*100,000}{341,672,244}=248.78[/tex]
Rounding up to the next whole unit, the projected incidence is 249 per 100,000.
Final answer:
To calculate the incidence rate of heart disease per 100,000 in the U.S. in 2020, divide the number of new cases (850,000) by the U.S. population (341,672,244), then multiply by 100,000, resulting in an incidence rate of approximately 248.7.
Explanation:
To calculate the projected incidence rate of heart disease per 100,000 in the United States in 2020, follow these steps:
First, determine the total number of new heart disease cases. Researchers estimated 850,000 new cases of heart disease will be recorded.
Second, determine the population of the United States at the midpoint of 2020, which is 341,672,244.
Finally, calculate the incidence rate using the formula: \((\frac{Number\ of\ new\ cases}{Population}) \times 100,000=Incidence\ Rate\ per\ 100,000\). Therefore, \((\frac{850,000}{341,672,244}) \times 100,000\) results in an incidence rate of approximately 248.7 new cases per 100,000 people.
Understanding this incidence rate can help in assessing the burden of heart disease on the U.S. population and guiding health policy decisions.
Bank of America's Consumer Spending Survey collected data on annualcredit card charges in seven different categories of expenditures:transportation, groceries, dining out, household expenses, homefurnishings, apparel, and entertainment (U.S. AirwaysAttache, December 2003). Using data from a sample of 42 creditcard accounts, assume that each account was used to identify theannual credit card charges for groceries (population 1) and theannual credit card charges for dining out (population 2). Using thedifference data, the sample mean difference was = $850, and the sample standard deviationwas sd = $1,123.
a.Formulate the null abd alternative hypothesis to test for no difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.
b.Use a .05 level of significance. Can you can conclude that the population mean differ? what is the p-value?
c. Which category, groceries or dining out, has a higher population mean annual credit card charge?What is the point estimate of the difference between the population means? What is the 95% confidence interval estimate of the difference between the population means?
Answer:
a) Null hypothesis: [tex]\mu_y- \mu_x = 0[/tex]
Alternative hypothesis: [tex]\mu_y -\mu_x \neq 0[/tex]
b) [tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{850 -0}{\frac{1123}{\sqrt{42}}}=4.905[/tex]
The next step is calculate the degrees of freedom given by:
[tex]df=n-1=42-1=41[/tex]
Now we can calculate the p value, since we have a two tailed test the p value is given by:
[tex]p_v =2*P(t_{(41)}>4.905) =7.6x10^{-6}[/tex]
So the p value is lower than any significance level given, so then we can conclude that we can to reject the null hypothesis that the difference between the two mean is equal to 0.
c) The confidence interval is given by:
[tex] \bar d \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
For this case we have a confidence of 1-0.05 = 0.95 so we need 0.05 of the area of the t distribution with 41 df on the tails. So we need 0.025 of the area on each tail, and the critical value would be:
[tex] t_{crit}= 2.02[/tex]
And if we find the interval we got:
[tex] 850- 2.02* \frac{1123}{\sqrt{42}}=499.96[/tex]
[tex] 850+ 2.02* \frac{1123}{\sqrt{42}}=1200.03[/tex]
We are confident 95% that the difference between the two means is between 499.96 and 1200.03
So we have enough evidence to conclude that one mean is higher than the other one, without conduct another hypothesis test because the confidence interval for the difference of means not contain the value of 0. And for this case the groceries would have a higher mean
Step-by-step explanation:
Previous concepts
A paired t-test is used to compare two population means where you have two samples in which observations in one sample can be paired with observations in the other sample. For example if we have Before-and-after observations (This problem) we can use it.
Part a
Let put some notation
x=test value for 1 , y = test value for 2
The system of hypothesis for this case are:
Null hypothesis: [tex]\mu_y- \mu_x = 0[/tex]
Alternative hypothesis: [tex]\mu_y -\mu_x \neq 0[/tex]
The first step is calculate the difference [tex]d_i=y_i-x_i[/tex] and we obtain this:
The second step is calculate the mean difference
[tex]\bar d= \frac{\sum_{i=1}^n d_i}{n}[/tex]
This value is given [tex] \bar d = 850[/tex]
The third step would be calculate the standard deviation for the differences.
This value is given also [tex] s_d = 1123[/tex]
Part b
The 4 step is calculate the statistic given by :
[tex]t=\frac{\bar d -0}{\frac{s_d}{\sqrt{n}}}=\frac{850 -0}{\frac{1123}{\sqrt{42}}}=4.905[/tex]
The next step is calculate the degrees of freedom given by:
[tex]df=n-1=42-1=41[/tex]
Now we can calculate the p value, since we have a two tailed test the p value is given by:
[tex]p_v =2*P(t_{(41)}>4.905) =7.6x10^{-6}[/tex]
So the p value is lower than any significance level given, so then we can conclude that we can to reject the null hypothesis that the difference between the two mean is equal to 0.
Part c
The confidence interval is given by:
[tex] \bar d \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
For this case we have a confidence of 1-0.05 = 0.95 so we need 0.05 of the area of the t distribution with 41 df on the tails. So we need 0.025 of the area on each tail, and the critical value would be:
[tex] t_{crit}= 2.02[/tex]
And if we find the interval we got:
[tex] 850- 2.02* \frac{1123}{\sqrt{42}}=499.96[/tex]
[tex] 850+ 2.02* \frac{1123}{\sqrt{42}}=1200.03[/tex]
We are confident 95% that the difference between the two means is between 499.96 and 1200.03
So we have enough evidence to conclude that one mean is higher than the other one, without conduct another hypothesis test because the confidence interval for the difference of means not contain the value of 0. And for this case the groceries would have a higher mean
Answer:
H_o : u_d = 0 , H_1 : u_d ≠ 0
H_o rejected , p < 0.01
[ 499.969 < d < 1200.301 ] , d = 850
Step-by-step explanation:
Given:
- Difference in mean d = 850
- Standard deviation s = 1123
- The sample size n = 42
- Significance level a = 0.05
Solution:
- Set up and Hypothesis for the difference in means test as follows:
H_o : Difference in mean u_d= 0
H_1 : Difference in mean u_d ≠ 0
- The t test statistics for hypothesis of matched samples is calculated by the following formula:
t = d / s*sqrt(n)
Hence,
t = 850 / 1123*sqrt(42)
t = 4.9053
Thus, the test statistics t = 4.9053.
- The p-value is the probability of obtaining the value of the test statistics or a value greater.
Using Table 2, of appendix B determine p with DOF = n - 1 = 42 - 1 = 41 , We get:
p < 2*0.05 ----> 0.01
Thus, p < 0.05 ....... Hence, H_o is rejected
- Set up and Hypothesis for the difference in means test as follows:
H_o : Difference in mean u_d =< 0
H_1 : Difference in mean u_d > 0
- The t test statistics for hypothesis of matched samples is calculated by te following formula:
t = d / s*sqrt(n)
Hence,
t = 850 / 1123*sqrt(42)
t = 4.9053
Thus, the test statistics t = 4.9053.
Using Table 2, of appendix B determine p with DOF = n - 1 = 42 - 1 = 41 , We get:
p < 0.005
Thus, p < 0.05 ....... Hence, H_o is rejected
Hence, the point estimate is d = $850
- The interval estimate of the difference between two population means is calculated by the following formula:
d +/- t_a/2*s / sqrt(n)
Where CI = 1 - a = 0.95 , a = 0.05 , a/2 = 0.025
Using Table 2, of appendix B determine p with DOF = n - 1 = 42 - 1 = 41 , We get:
t_a/2 = t_0.025 = 2.020
Therefore,
d - t_a/2*s / sqrt(n)
850 - 2.020*1123 / sqrt(20)
= 499.969
And,
d + t_a/2*s / sqrt(n)
850 + 2.020*1123 / sqrt(20)
= 1200.031
- The 95% CI of the difference between two population means is:
[ 499.969 < d < 1200.301 ]
If a = 6 and c = 15, what is the measure of ∠A? (round to the nearest tenth of a degree) Q: A: A) 21.8° B) 22.7° C) 23.6° D) 66.4°
Answer:
Option C) 23.6°
Step-by-step explanation:
we know that
In this problem the triangle ABC is a right triangle
see the attached figure to better understand the problem
[tex]sin(A)=\frac{BC}{AB}[/tex] ----> by SOH (opposite side divided by the hypotenuse)
substitute the given values
[tex]sin(A)=\frac{6}{15}[/tex]
using a calculator
[tex]A=sin^{-1}(\frac{6}{15})=23.6^o[/tex]
Suppose that you play the game with three different friends separately with the following results: Friend A chose scissors 100 times out of 400 games, Friend B chose scissors 20 times out of 120 games, and Friend C chose scissors 65 times out of 300 games. Suppose that for each friend you want to test whether the long-run proportion that the friend will pick scissors is less than 1/3.
1) Select the appropriate standardized statistics for each friend from the null distribution produced by applet.
-3.47 (100 out of 400; 25%), -4.17 (20 out of 120; 16.7%), -3.80 (65 out of 300; 21.7%)
-3.80 (100 out of 400; 25%), -3.47 (20 out of 120; 16.7%), -4.17 (65 out of 300; 21.7%)
-3.47 (100 out of 400; 25%), -3.80 (20 out of 120; 16.7%), -4.17 (65 out of 300; 21.7%)
-4.17 (100 out of 400; 25%), -3.80 (20 out of 120; 16.7%), -3.47 (65 out of 300; 21.7%)
Answer:
Friend A
[tex]\hat p_A= \frac{100}{400}=0.25[/tex]
[tex]z=\frac{0.25 -0.333}{\sqrt{\frac{0.333(1-0.333)}{400}}}\approx -3.47[/tex]
Friend B
[tex]\hat p_B= \frac{20}{120}=0.167[/tex]
[tex]z=\frac{0.167 -0.333}{\sqrt{\frac{0.333(1-0.333)}{120}}}\approx -3.80[/tex]
Friend C
[tex]\hat p_C= \frac{65}{300}=0.217[/tex]
[tex]z=\frac{0.217-0.333}{\sqrt{\frac{0.333(1-0.333)}{300}}}\approx -4.17[/tex]
So then the best solution for this case would be:
-3.47 (100 out of 400; 25%), -3.80 (20 out of 120; 16.7%), -4.17 (65 out of 300; 21.7%)
Step-by-step explanation:
Data given and notation
n represent the random sample taken
X represent the number of scissors selected for each friend
[tex]\hat p=\frac{X}{n}[/tex] estimated proportion of scissors selected for each friend
[tex]p_o=\frac{1}{3}=0.333[/tex] is the value that we want to test
[tex]\alpha[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the proportion that the friend will pick scissors is less than 1/3 or 0.333, the system of hypothesis would be:
Null hypothesis:[tex]p\geq 0.333[/tex]
Alternative hypothesis:[tex]p < 0.333[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
Friend A
[tex]\hat p_A= \frac{100}{400}=0.25[/tex]
[tex]z=\frac{0.25 -0.333}{\sqrt{\frac{0.333(1-0.333)}{400}}}\approx -3.47[/tex]
Friend B
[tex]\hat p_B= \frac{20}{120}=0.167[/tex]
[tex]z=\frac{0.167 -0.333}{\sqrt{\frac{0.333(1-0.333)}{120}}}\approx -3.80[/tex]
Friend C
[tex]\hat p_C= \frac{65}{300}=0.217[/tex]
[tex]z=\frac{0.217-0.333}{\sqrt{\frac{0.333(1-0.333)}{300}}}\approx -4.17[/tex]
So then the best solution for this case would be:
-3.47 (100 out of 400; 25%), -3.80 (20 out of 120; 16.7%), -4.17 (65 out of 300; 21.7%)
A circle's radius that has an initial radius of 0 cm is increasing at a constant rate of 5 cm per second. a. Write a formula to expresses the radius of the circle, r (in cm), in terms of the number of seconds, t since the circle started growing. Preview b. Write a formula to express the area of the circle, A (in square cm), in terms of the circle's radius, r (in cm). A = Preview c. Write a formula to expresses the circle's area, A (in square cm), in terms of the number of seconds, t, since the circle started growing. A = Preview d. Write your answer to part (c) in expanded form - so that your answer does not contain parentheses.
Answer:
a. r = 5t
b. [tex]A = \pi r^2[/tex]
c. [tex]A = \pi (5t)^2[/tex]
d. [tex]A = 25\pi t^2[/tex]
Step-by-step explanation:
a. Since the radius is increasing at a constant rate of 5 cm per second.
r = 5t
where r is the radius at time t (seconds)
b. Area of circle [tex]A = \pi r^2[/tex]
c. We can substitute r = 5t into the area formula to have
[tex]A = \pi r^2 = \pi (5t)^2[/tex]
d. In expand form
[tex]A = \pi (5t)^2 = 25\pi t^2[/tex]
a. The expression is R = 5t
b. The area of the circle in terms of R is A = πR²
c. The area of the circle in terms of t is A = π(5t)²
d. The area of the circle in terms of t in the expanded form is A = 25π×t²
Linear systemIt is a system of an equation in which the highest power of the variable is always 1. A one-dimension figure that has no width. It is a combination of infinite points side by side.
CircleIt is a locus of a point drawn at an equidistant from the center. The distance from the center to the circumference is called the radius of the circle.
Given
R = 5t where R is the radius, and t be the time.
Thus, the answer will be
a. The expression will be
R = 5t
b. The area of the circle in terms of R will be
Area = πR²
c. The area of the circle in terms of t will be
Area = πR²
Area = π(5t)²
d. The area of the circle in terms of t in the expanded form will be
Area = πR²
Area = π(5t)²
Area = 25π×t²
More about the linear system link is below.
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There are 39 members on the Central High School student government council. When a vote took place on a certain proposal, all of the seniors and none of the freshmen voted for it. Some of the juniors and some of the sophomores voted for the proposal and some voted against it.If a simple majority of the votes cast is required for the proposal to be adopted, which of the following statements, if true, would enable you to determine whether the proposal was adopted?a. There are more seniors than freshmen on the council.b. A majority of the freshmen and a majority of the sophomores voted for the proposal. c. There are 18 seniors on the council.d. There are the same number of seniors and freshmen combined as there are sophomores and juniors combined.e. There are more juniors than sophomores and freshmen combined, and more than 90% of the juniors voted against the proposal.
Answer:
Option b.
Step-by-step explanation:
Statement b would be true in this case.
Let's gather data from the question:
student council = seniors + juniors
Now, some few things to note:
1. Senior students are in their 12th grade. This is the senior year in high school.
2. The sophomore is the 10th year in school. These are not senior year students.
Isolating the students, the sophomore + junior students are likely to be the majority here.
Some junior and sophomore students voted for the proposal so it means that the combined number will be: all senior students + some juniors + some sophomores.
Therefore, the majority of the freshmen and a majority of the sophomores voted for the proposal.
Find the equation for the plane through the points Po(3,-2,5), Qo(-3,-1,-5), and Ro(0,-4,4) The equation of the plane is Type an equation.)
Answer:
- 21 x + 24 y + 15 z =120
Step-by-step explanation:
Given that
Po(3,-2,5), Qo (-3,-1,-5), and Ro (0,-4,4) ,These are the point in the space.
We know that equation of a plane is given as
[tex]\begin{vmatrix}x-x_1 & y-y_1 &z-z_1 \\ x_2-x_1 & y_2-y_1 &z_2-z_1 \\ x_3-x_1 &y_3-y_1 & z_3-z_1\end{vmatrix}=0\\[/tex]
[tex]\begin{vmatrix}x-0 & y+4 &z-4 \\ 3-0 & -2+4 &5-4 \\ -3-0 &-1+4 & -5-4\end{vmatrix}=0.[/tex]
[tex]\begin{vmatrix}x & y+4 &z-4 \\ 3 & 2 &1 \\ -3 &3 & -9\end{vmatrix}=0.[/tex]
Now by solving above determinate we get
x( -18 -3 ) -(y+4 ) ( -27 +3 ) + ( z- 4) (9+6) = 0
-21 x +24 y -24 x 4 + 15 z - 24 = 0
- 21 x + 24 y + 15 z -120 = 0
- 21 x + 24 y + 15 z =120
Therefore the equation of the plane will be
- 21 x + 24 y + 15 z =120
A 22 KHz baseband channel is used by a digital transmission system. Suppose ideal pulses are sent at the Nyquist rate, and the pulses can take 1024 levels. There is no noise in the system. What is the bit rate of this system
Answer:
Bit rate = 440 kBits/sec
Step-by-step explanation:
Band width = W= 22 kHz
Number of levels = L = 1024 levels
Bit per sample:
[tex]m=log_2 L\\\\m =log_2(1024)\\\\m=10 bits/sample[/tex]
Ideal pulses are sent at the Nyquist rate then bit rate = 2 x W x m
[tex]bit\,\,rate= 2\times 22\times 10^3\times 10\\\\bit\,\,rate= 440\times 10^3 bits\,sec^{-1}[/tex]
bit rate = 440 kBits/sec
A financial talk show host claims to have a 55.3 % success rate in his investment recommendations. You collect some data over the next few weeks, and find that out 10 recommendations, he was correct 3 times. If the claim is correct and the performance of recommendations is independent, what is the probability that you would have observed 4 or fewer successfu
Answer:
There is a 25.52% probability of observating 4 our fewer succesful recommendations.
Step-by-step explanation:
For each recommendation, there are only two possible outcomes. Either it was a success, or it was a failure. So we use the binomial probability distribution to solve this problem.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this problem we have that:
[tex]p = 0.553, n = 10[/tex]
If the claim is correct and the performance of recommendations is independent, what is the probability that you would have observed 4 or fewer successful:
This is
[tex]P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{10,0}.(0.553)^{0}.(0.447)^{10} = 0.0003[/tex]
[tex]P(X = 1) = C_{10,1}.(0.553)^{1}.(0.447)^{9} = 0.0039[/tex]
[tex]P(X = 2) = C_{10,2}.(0.553)^{2}.(0.447)^{8} = 0.0219[/tex]
[tex]P(X = 3) = C_{10,3}.(0.553)^{3}.(0.447)^{7} = 0.0724[/tex]
[tex]P(X = 4) = C_{10,4}.(0.553)^{4}.(0.447)^{6} = 0.1567[/tex]
[tex]P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.0003 + 0.0039 + 0.0219 + 0.0724 + 0.1567 = 0.2552[/tex]
There is a 25.52% probability of observating 4 our fewer succesful recommendations.
A sports manufacturer produces two products: footballs and baseballs. These products can be produced either during the morning shift or the evening shift. The cost of manufacturing the football and the baseball in the morning shift is $20 each, and the cost of manufacturing the football and the baseball in the evening shift is $25 each. The amounts of labor, leather, inner plastic lining, and demand requirements are given as follows: Resource Football Baseball Labor (hours/unit) 0.75 2 Leather (pounds/unit) 7 15 Inner plastic lining (pounds/unit) 0.5 2 Total demand (units) 1500 1200 Based on the information about the company, we know that the maximum labor hours available in the morning shift and evening shift are 5,000 hours and 2,000 hours, respectively, per month. The maximum amount of leather available for the morning shift is 15,000 pounds per month and 14,000 pounds per month for the evening shift. The maximum amount of inner plastic lining available for the morning shift is 2,000 pounds per month and 1,500 pounds per month for the evening shift.
Step-by-step explanation:
From the above illlustration,
Let x, be the number of footballs produced in the morning shift,
y, the number of baseball in the morning shift,
z, the number of football in the evening shift,
t, the number of baseball in the evening shift.
Minimizing the objective function,
min {20(x+y) + 25(z + t)}
Therefore, since the number of labor hours is for both shifts(morning and evening shifts), we add the following constraints:
0.75x + 2y ≤ 5000
0.75z + 2t ≤ 2000
Remember, the amount of leather available in the shifts is also limited. The following constraints are got:
7x + 15y ≤ 15000
7z + 15t ≤1 4000
Also, adding the constraints for the use of inner plastic lining, we have:
0.5x + 2y ≤ 2000
0.5z + 2t ≤ 1500
Modelling their demands through the following constraints:
x + z ≥ 1500
y + t ≥ 1200
Also, we are producing whole number of baseballs or footballs but we only, so
x, y, z, t ∈Z.
Finally,
min20(x + y) + 25(z + t)
0.75x + 2y ≤ 5000
0.75z + 2t ≤ 2000
7x + 15y ≤ 15000
7z + 15t ≤ 14000
0.5x + 2y ≤ 2000
0.5z + 2t ≤ 1500
x + z ≥ 1500
y + t ≥ 1200
y, x, t, z ∈ Z.
The product of Donnie's height and
8
is
128
.
Answer:
128
Step-by-step explanation:
Please help! I dont know how to figure this out.
Answer: the third option is the correct answer.
Step-by-step explanation:
Looking at the line plot,
There are 3 bags of oranges that weigh 3 7/8 pounds each. Converting 3 7/8 to improper fraction, it becomes 31/8 pounds. Therefore, the weight of the three bags would be
3 × 31/8 = 93/8 pounds
There are 2 bags of oranges that weigh 4 pounds each. Therefore, the weight of the four bags would be
2 × 4 = 8 pounds
There are 3 bags of oranges that weigh 4 1/8 pounds each. Converting 4 1/8 to improper fraction, it becomes 33/8 pounds. Therefore, the weight of the three bags would be
3 × 33/8 = 99/8 pounds
There are 2 bags of oranges that weigh 4 2/8 pounds each. Converting 4 2/8 to improper fraction, it becomes 34/8 pounds. Therefore, the weight of the three bags would be
2 × 34/8 = 68/8
Therefore, the total number of oranges would be
93/8 + 33/8 + 4 + 102/8 = (93 + 64 + 99 + 68)/8 = 324/8 = 40 1/2 pounds
Be my Valentine: The following frequency distribution presents the amounts, in dollars, spent for Valentine's Day gifts in a survey of 120 U.S. adults in a recent year. Approximate the mean amount spent on Valentine's Day gifts to two decimal places.Amount Frequency0-19.99 16 20.00-39.99 1340.00-59.99 21 60.00-79.99 19 80.00-99.99 12 100.00-119.99 10 120.00-139.99 7140.00-159.99 8160.00-179.99 7180,00-199.99 1200.00-219.99 3220.00-239.99 2240.00-259.99 1
Answer:
the mean is 82.75
Step-by-step explanation:
Amount Frequency Mid Point fx
0-19.99 16 9.995 159.92
20.00-39.99 13 29.995 389.935
40.00-59.99 21 49.995 1049.895
60.00-79.99 19 69.995 1329.905
80.00-99.99 12 89.995 1079.94
100.00-119.99 10 109.995 1099.95
120.00-139.99 7 129.995 909.965
140.00-159.99 8 149.995 1199.96
160.00-179.99 7 169.995 1189.965
180,00-199.99 1 189.995 189.995
200.00-219.99 3 209.995 629. 985
220.00-239.99 2 229.995 459.99
240.00-259.99 1 249.995 240.995
∑ 120 9930.4
Mean = ∑fx/∑x
Mean = 9930.4/120=82.7533 = 82.75
Answer: The mean amount spent on Valentine's day is;
$82.83
Step-by-step explanation: To find the mean amount we first arrange the numbers in a frequency table, then solve.
STEP1:
AMOUNT FREQUENCY
0-19.99 16
20.00-39.99 13
40.00-59.99 21
60.00-79.99 19
80.00-99.99 12
100.00-119.99 10
120.00-139.99 7
140.00-159.99 8
160.00-179.99 7
180,00-199.99 1
200.00-219.99 3
220.00-239.99 2
240.00-259.99 1
STEP 2: Find the center of each amount, to do this we have to find the average value of the amounts.
For the first amount is;
(0+19.99)/2 = 9.995
For the second amount is;
(20+39.99)/2 =29.995
Solving this for all the amount. Therefore the table comes
AMOUNT FREQUENCY
9.995 16
29.995 13
49.995 21
69.995 19
89.995 12
109.995 10
129.995 7
149.995 8
169.995 7
189.995 1
209.995 3
229.995 2
249.995 1
STEP 3: multiple each amount in step 2 with the frequency.
For the first amount;
9.995×16 = 159.92
For the second amount;
29.995×13= 389.935
For the third amount;
49.995×21= 1049.895
For the fourth amount;
69.99×19= 1329.81
For the fifth amount;
89.995×12=1079.94
For the six amount;
109.995×10= 1099.95
For the sixth amount;
129.995×7= 909.965
For the seventh amount;
149.995×8= 1199.96
For the eight amount;
169.995×7= 1189.965
For the ninth amount;
189.995×1= 189.995
For the tenth amount;
209.995×3= 629.985
For the eleventh amount;
229.995×2=459.99
For the twelveth amount;
249.995×1= 249.995
STEP 4: Sum up all the answers from the multiplication in step 3
Therefore;
159.92+389.935+1049.895+1329.81+1079.94+1099.95+909.965+1199.96+1189.965+189.995+629.985+459.99+629.985+459.99+249.995 = 9939.995
STEP 5: divide the sum of the value seen in step 4 with the total number of frequency to get the mean value.
The total number of frequency is 120
Therefore;
9939.305÷120=82.827541
Take the value to two decimal place, it becomes;
$82.83 this is the mean value of money spent on Valentine's day.
An experiment results in one of three mutually exclusive events, A,B,C. it is known that p(A) =.30, p(b) =.55 and p(c) =.15.
A. find each of the following probabilities.1. P(AUB)2. P(A∩C)3. P(A|B)4. P(BUC)B. Are B and C Independent Events? Explain.
Answer:
A. 1. P(A∪B)=0.85
2. P(A∩C)=0.045
3. P(A/B)=0.3
4. P(B∪C)=0.70
B. Event B and Event C are dependent
Step-by-step explanation:
A. As events are mutually exclusive, so,
P(A∪B)=P(A)+P(B)
P(A∩B)=P(A)*P(B)
1. P(A∪B)=?
P(A∪B)=P(A)+P(B)=0.3+0.55=0.85
P(A∪B)=0.85
2. P(A∩C)
P(A∩C)=P(A)*P(C)=0.30*0.15=0.045
P(A∩C)=0.045
3. P(A/B)
P(A/B)=P(A∩B)/P(B)
P(A∩B)=P(A)*P(B)=0.30*0.55=0.165
P(A/B)=P(A∩B)/P(B)=0.165/0.55=0.3
P(A/B)=0.3
4. P(B∪C)
P(B∪C)=P(B)+P(C)=0.55+0.15=0.70
P(B∪C)=0.70
B.
The event B and C are mutually exclusive and events B and event C are dependent i.e. P(B and C)≠P(B)P(C)
The events are mutually exclusive i.e. P(B and C)=0
whereas P(B)*P(C)=0.55*0.15=0.0825
Mutually exclusive events are independent only if either one of two or both events has zero probability of occurring.
Thus, event B and C are dependent
A) 1:P(A∪B)=0.85
2: P(A∩C)=0.045
3: P(A/B)=0.3
4: P(B∪C)=0.70
B) Events B and C are dependent events.
Since all three events are mutually exclusive:
So, P(A∪B)=P(A)+P(B)
P(A∪B) = 0.30+0.55
P(A∪B) = 0.85
P(A∩B)=P(A)P(B)
P(A∩B) = 0.30*0.55 = 0.165
P(A/B)=P(A∩B)/P(B)
P(A/B) = 0.165/0.55 = 0.3
Similarly, P(A∩C) =0.045
P(BUC) = 0.70
Events B and C are dependent events because they will be independent only if there is zero possibility of their occurrence.
Therefore, A) 1:P(A∪B)=0.85
2: P(A∩C)=0.045
3: P(A/B)=0.3
4: P(B∪C)=0.70
B) Events B and C are dependent events.
To get more about probability visit:
https://brainly.com/question/251701