The probability of A = 1/4 = 0.25
The probability of B = 4/5 = 0.80
P(A ∩ B) = P(A) * P (B) = (0.25) * (0.80) = 0.20
P(A ∩ B') = P(A) * P (B') = (0.25) * (1-0.80) = 0.05
P(A' ∩ B') = P(A') * P (B') = (1-0.25) * (1-0.80) = 0.15
Answer:
A. P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.25 + 0.80 - 0.20 = 0.85
B. P(A U B') = P(A) + (1 - P(B)) - P(A ∩ B') = 0.25 + (1-0.80) - 0.05 = 0.40
C. P(A' U B') = (1 - P(A)) + (1 - P(B)) - P(A' ∩ B') = (1-0.25) + (1-0.80) - 0.15 = 0.80
The probabilities using the addition rule of probabilities are P(A ∪ B) = 17/20, P(A ∪ B') = 2/5, and P(A' ∪ B') = 4/5.
In solving this problem, we will use the addition rule for probabilities to find the likelihood of different combinations of randomly generated web ad designs based on given design elements.
A. P(A ∪ B)
We'll start by finding the probability of event A (the design color is red) and B (the font size is not the smallest one) occurring independently. Since there are four different colors, the probability of A is 1/4. When it comes to event B, since there are five different font sizes and the event is that the font size is not the smallest one, there are four font sizes that meet the criteria.
Therefore, the probability of B is 4/5. We can now calculate P(A ∪ B) using the formula P(A) + P(B) - P(A AND B). Assuming that the color choice and font size are independent of each other (the choice of one does not affect the choice of the other), the probability of both A and B occurring (A AND B) is simply the product of their separate probabilities: P(A) x P(B) = (1/4) x (4/5) = 1/5.
So, P(A ∪ B) = P(A) + P(B) - P(A AND B) = (1/4) + (4/5) - (1/5) = 1/4 + 16/20 - 4/20 = 1/4 + 12/20 = 5/20 + 12/20 = 17/20.
B. P(A ∪ B')
To find P(A ∪ B'), we first need to calculate P(B'). Since the probability of B is 4/5, the probability of B' (the font size is the smallest one) is the remaining fraction of 1, which is 1/5. We then apply the addition rule, which becomes P(A \∪\ B') = P(A) + P(B') - P(A AND B'), with P(A AND B') being the probability of choosing the red color and the smallest font size. Because A and B' are independent, P(A AND B') = P(A) x P(B') = (1/4) x (1/5) = 1/20.
Therefore, P(A ∪ B') = P(A) + P(B') - P(A AND B') = 1/4 + 1/5 - 1/20 = 5/20 + 4/20 - 1/20 = 8/20 or 2/5.
C. P(A' ∪ B')
To find P(A' ∪ B'), we need to determine the probabilities of A' and B' occurring separately. P(A') is the probability of not choosing the red color, which is the complement of P(A), meaning P(A') = 1 - P(A) = 3/4. P(B') as we calculated before, is 1/5. As these events are independent, P(A' AND B') = P(A') x P(B') = (3/4) x (1/5) = 3/20.
Thus, P(A' ∪ B') = P(A') + P(B') - P(A' AND B') = 3/4 + 1/5 - 3/20 = 15/20 + 4/20 - 3/20 = 16/20 or 4/5.
What is the surface area of the figure?
144π cm²
12π cm²
36π cm²
24π cm²
Answer:
36π cm^2.
Step-by-step explanation:
This is a sphere . Surface area = 4πr^2.
This sphere has surface area = 4π3^2
= 36π.
The surface area of the sphere would be = 36πcm². That is option C.
What is area ?Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.
here, we have,
to calculate the surface area of a sphere:
The surface area of a sphere can be calculated through the use of the formula = 4πr²
Where,
radius (r) = 3 cm
surface area
=4πr²
= 4π × 3²
= 36π cm² ( in the terms of π)
Hence, The surface area of the sphere would be = 36πcm². That is option C.
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Which statement best describes how to determine whether f(x) = 9 – 4x2 is an odd function?
A. Determine whether 9 – 4(–x)2 is equivalent to 9 – 4x2.
B. Determine whether 9 – 4(–x2) is equivalent to 9 + 4x2.
C. Determine whether 9 – 4(–x)2 is equivalent to –(9 – 4x2).
D. Determine whether 9 – 4(–x2) is equivalent to –(9 + 4x2).
Answer:
Option C - determine whether [tex]9-4(-x)^2[/tex] is equivalent to [tex]-(9-4x^2)[/tex] or not.
Step-by-step explanation:
To find : Which statement best describes how to determine whether [tex]f(x) = 9-4x^2[/tex] is an odd function?
Solution :
We have a property for odd functions,
Let f(x) be an odd function then it must satisfy
[tex]f(-x)= -f(x)[/tex]
Now, we have been given the function [tex]f(x) = 9-4x^2[/tex]
For this function to be odd, it must satisfy the above property.
Replace x with -x,
[tex]f(-x)=9-4(-x)^2[/tex]
and
[tex]-f(x)=-(9-4x^2)[/tex]
Hence, in order to the given function to be an odd function, we must determine whether [tex]9-4(-x)^2[/tex] is equivalent to [tex]-(9-4x^2)[/tex] or not.
Therefore, C is the correct option.
In one U.S. city, the quadratic function f (x )equals 0.0039 x squared minus 0.42 x plus 36.79 models the median, or average, age, y, at which men were first married x years after 1900. In which year was this average age at a minimum (round to the nearest year)? What was the average age at first marriage for that year (round to the nearest tenth)?
Answer:
The average age was minimum at 1954 and the average age is 25.5.
Step-by-step explanation:
The given quadratic function is
[tex]f(x)=0.0039x^2-0.42x+36.79[/tex]
It models the median, or average, age, y, at which men were first married x years after 1900.
In the above equation leading coefficient is positive, so it is an upward parabola and vertex of an upward parabola, is point of minima.
We need to find the year in which the average age was at a minimum.
If a quadratic polynomial is [tex]f(x)=ax^2+bx+c[/tex], then vertex is
[tex]Vertex=(-\dfrac{b}{2a},f(-\dfrac{b}{2a}))[/tex]
[tex]-\dfrac{b}{2a}=-\dfrac{(-0.42)}{2(0.0039)}=53.846153\approx 54[/tex]
54 years after 1900 is
[tex]1900+54=1954[/tex]
Substitute x=54 in the given function.
[tex]f(54)=0.0039(54)^2-0.42(54)+36.79=25.4824\approx 25.5[/tex]
Therefore, the average age was minimum at 1954 and the average age is 25.5.
The year when the average age at first marriage was at a minimum in a specific U.S. city was approximately 1954. The average age at first marriage for that year was approximately 28.4 years.
Explanation:To find the year when the average age at first marriage was at a minimum, we need to determine the x-value at the vertex of the quadratic function. The x-value at the vertex can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic function. For the given function f(x) = 0.0039x2 - 0.42x + 36.79, the x-value at the vertex is x = -(-0.42)/(2*0.0039) = 53.85. Since the x-value represents years after 1900, we add 1900 to get the year: 1900 + 53.85 ≈ 1954.
To find the average age at first marriage for that year, we substitute x = 53.85 into the quadratic function. f(53.85) = 0.0039(53.85)2 - 0.42(53.85) + 36.79 ≈ 28.4. Therefore, the average age at first marriage for the year 1954 was approximately 28.4 years.
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Marcus sold brownies at a bake sale. He sold d dollars worth of brownies he spent a total of $5.50 on materials, so his total profit p in dollars can be found by subtracting $5.50 from his earnings. Write an equation that represents this situation
Answer:
d = $5.50 - p
Step-by-step explanation:
Answer: d = $5.50 - p
Step-by-step explanation:
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Discrete or Continous?
A) the number of passengers in a passenger vehicle on a highway at rush hour
B) the air pressure of a tire on an automobile
C) the weight of refuse on a truck arriving at a landfill
Final answer:
Data can be categorized as discrete or continuous. Discrete data consist of countable values, while continuous data can take on any value within a range and are measurable. Examples include the number of passengers (discrete) and air pressure of a tire (continuous).
Explanation:
When categorizing data, it's important to determine whether the data are discrete or continuous. A discrete variable is one that can only take on certain, typically countable, values. Continuous variables, on the other hand, can take on any value within a range and are measurable.
Examples:
The number of passengers in a passenger vehicle on a highway at rush hour is discrete, as you can count passengers.The air pressure of a tire on an automobile is continuous, as pressure can be measured and can vary along a continuum within a range.The weight of refuse on a truck arriving at a landfill is continuous because weight can take on any value within a range and is not countable.Further examples:
The number of gallons of gasoline necessary to fill an automobile gas tank is discrete.The number of cm in 2 m is discrete, as centimeters can be counted and there is a fixed number of them in 2 meters.The mass of a textbook is continuous, as mass can vary along a continuum and is measured.The time required to drive from San Francisco to Kansas City at an average speed of 53 mi/h is continuous, because time can take any value and is measured.The temperature outside is 15 degrees Fahrenheit . If the temperature drops 20 degrees , will the outside temperature be represented by a positive integer or negative integer?Explain your reasoning
Answer:
The temperature would be -5 degrees Fahrenheit
Step-by-step explanation: It's represented by a negative integer because 15 - 20 = -5. This means the temperature outside would be -5 degrees Fahrenheit.
Hope this helps! (:
The temperature would be -5 degrees Fahrenheit if The temperature outside is 15 degrees Fahrenheit . If the temperature drops 20 degrees,
What is unit conversion?It is defined as the conversion from one quantity unit to another quantity unit followed by the process of division, multiplication by a conversion factor.
It is given that:
The temperature outside is 15 degrees Fahrenheit . If the temperature drops 20 degrees
=15 - 20
= -5.
A negative sign means the temperature outside would be -5 degrees Fahrenheit.
Thus, the temperature would be -5 degrees Fahrenheit if The temperature outside is 15 degrees Fahrenheit. If the temperature drops 20 degrees,
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One night a theater sold 548 movie tickets. An adult's costs $6.50 an child's cost $3.50. In all, $2,881 was takin in. How many of each kind of tickets were sold?
Answer:
321 adult227 childStep-by-step explanation:
The fraction of tickets that are adult tickets is ...
((average price per ticket) - (child's ticket cost)) / (difference in ticket costs)
so the fraction of adult tickets is ...
((2881/548) -3.50)/(6.50 -3.50) = 321/548
Then the number of adult tickets is ...
(321/548)·548 = 321
and the number of child tickets is ...
548 -321 = 227
321 adult and 227 child tickets were sold that night.
_____
If you want to write an equation, you can let "a" represent the number of adult tickets sold. Total revenue is ...
6.50a +3.50(548 -a) = 2881
3.00a +1918 = 2881 . . . . . . eliminate parentheses
3a = 963 . . . . . . . . . . . . . . . subtract 1918
a = 321 . . . . . . . . . . . . . . . . . divide by 3
The number of child tickets is ...
548 -a = 548 -321 = 227
100 points , please help. I am not sure if I did this correct if anyone can double-check me thanks!
my answer:
2. In order to find the definite integral of the riemann sum given to us. We need to label everything out. We know that our delta x = 3/n , a=1 and that b=4. We found B by subtracting
b-a=delta x
b-1=3
b=4.
Then now we plug everything in giving us our final answer, ⎰^4 and 1 on the bottom (sqrt 1 + 3/n) dx.
Step-by-step explanation:
[tex]\lim_{n \to \infty} \sum\limits_{k=1}^{n}f(x_{k}) \Delta x = \int\limits^a_b {f(x)} \, dx \\where\ \Delta x = \frac{b-a}{n} \ and\ x_{k}=a+\Delta x \times k[/tex]
In this case we have:
Δx = 3/n
b − a = 3
a = 1
b = 4
So the integral is:
∫₁⁴ √x dx
To evaluate the integral, we write the radical as an exponent.
∫₁⁴ x^½ dx
= ⅔ x^³/₂ + C |₁⁴
= (⅔ 4^³/₂ + C) − (⅔ 1^³/₂ + C)
= ⅔ (8) + C − ⅔ − C
= 14/3
If ∫₁⁴ f(x) dx = e⁴ − e, then:
∫₁⁴ (2f(x) − 1) dx
= 2 ∫₁⁴ f(x) dx − ∫₁⁴ dx
= 2 (e⁴ − e) − (x + C) |₁⁴
= 2e⁴ − 2e − 3
∫ sec²(x/k) dx
k ∫ 1/k sec²(x/k) dx
k tan(x/k) + C
Evaluating between x=0 and x=π/2:
k tan(π/(2k)) + C − (k tan(0) + C)
k tan(π/(2k))
Setting this equal to k:
k tan(π/(2k)) = k
tan(π/(2k)) = 1
π/(2k) = π/4
1/(2k) = 1/4
2k = 4
k = 2
The distribution of scores on the SAT is approximately normal with a mean of mu = 500 and a standard deviation of sigma = 100. For the population of students who have taken the SAT, a.What proportion have SAT scores greater than 700? b.What proportion have SAT scores greater than 550? c.What is the minimum SAT score needed to be in the highest 10% of the population? d.If the state college only accepts students from the top 60% of the SAT distribution, what is the minimum SAT score needed to be accepted ?
Answer:
a. 2.28%
b. 30.85%
c. 628.16
d. 474.67
Step-by-step explanation:
For a given value x, the related z-score is computed as z = (x-500)/100.
a. The z-score related to 700 is (700-500)/100 = 2, and P(Z > 2) = 0.0228 (2.28%)
b. The z-score related to 550 is (550-500)/100 = 0.5, and P(Z > 0.5) = 0.3085 (30.85%)
c. We are looking for a value b such that P(Z > b) = 0.1, i.e., b is the 90th quantile of the standard normal distribution, so, b = 1.281552. Therefore, P((X-500)/100 > 1.281552) = 0.1, equivalently P(X > 500 + 100(1.281552)) = 0.1 and the minimun SAT score needed to be in the highest 10% of the population is 628.1552
d. We are looking for a value c such that P(Z > c) = 0.6, i.e., c is the 40th quantile of the standard normal distribution, so, c = -0.2533471. Therefore, P((X-500)/100 > -0.2533471) = 0.6, equivalently P(X > 500 + 100(-0.2533471)), and the minimun SAT score needed to be accepted is 474.6653
Using the normal distribution, it is found that:
a) 0.0228 = 2.28% of students have SAT scores greater than 700.
b) 0.3085 = 30.85% of students have SAT scores greater than 550.
c) The minimum SAT score needed to be in the highest 10% of the population is 628.
d) The minimum SAT score needed to be accepted is 475.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.In this problem:
The mean is [tex]\mu = 500[/tex]The standard deviation is [tex]\sigma = 100[/tex].Item a:
This proportion is 1 subtracted by the p-value of Z when X = 700, thus:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{700 - 500}{100}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a p-value of 0.9772.
1 - 0.9772 = 0.0228.
0.0228 = 2.28% of students have SAT scores greater than 700.
Item b:
This proportion is 1 subtracted by the p-value of Z when X = 550, thus:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{550 - 500}{100}[/tex]
[tex]Z = 0.5[/tex]
[tex]Z = 0.5[/tex] has a p-value of 0.6915.
1 - 0.6915 = 0.3085.
0.3085 = 30.85% of students have SAT scores greater than 550.
Item c:
This is the 100 - 10 = 90th percentile, which is X when Z has a p-value of 0.9, so X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 500}{100}[/tex]
[tex]X - 500 = 1.28(100)[/tex]
[tex]X = 628[/tex]
The minimum SAT score needed to be in the highest 10% of the population is 628.
Item d:
This is the 100 - 60 = 40th percentile, which is X when Z has a p-value of 0.4, so X when Z = -0.25.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.25 = \frac{X - 500}{100}[/tex]
[tex]X - 500 = -0.25(100)[/tex]
[tex]X = 475[/tex]
The minimum SAT score needed to be accepted is 475.
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The population of grand island, nebraska, grew by 600,000 people between 1995 and 2005, one fifth more than the town council originally predicted the city's population would grow by ?
Answer:
500000 people
Step-by-step explanation:
The population grew by 600,000 which is 120% the earlier prediction by the town council.
Using direct proportion
600,000 -------- 120%
X --------- 100%
X = (600000 × 100) ÷ 120 = 500000
Therefore the earlier prediction by the town council is 500000 people
The student's question is a mathematical problem calculating population growth predictions. The town council of Grand Island originally predicted a growth of 500,000 people, which is 20% less than the actual growth of 600,000 people.
To determine the prediction made by the town council, we can use the fact that the actual growth exceeded the prediction by one fifth (or 20%). If the actual growth was 600,000 people, the predicted growth can be calculated by dividing 600,000 by 1.2, as the actual growth represents 120% of the predicted value (100% original prediction + 20% excess).
Calculating the Predicted Population Growth
To find the town council's predicted growth, we can set up the equation:
Actual Growth = Predicted Growth + (Predicted Growth × 0.20)600,000 = Predicted Growth × 1.20Predicted Growth = 600,000 / 1.20Predicted Growth = 500,000Therefore, the town council had originally predicted that the population of Grand Island, Nebraska, would increase by 500,000 people between 1995 and 2005.
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Find m∠R.
Write your answer as an integer or as a decimal rounded to the nearest tenth.
m∠R = °
Answer:
[tex]m\angle R=69.4^o[/tex]
Step-by-step explanation:
we know that
In the right triangle PQR
[tex]tan(R)=\frac{PQ}{QR}[/tex] ----> by TOA (opposite side divided by adjacent side)
substitute the given values
[tex]tan(R)=\frac{8}{3}[/tex]
using a calculator
[tex]m\angle R=tan^{-1}(\frac{8}{3})=69.4^o[/tex]
what does 124.06=2.35h+72.36
a.16
b.22
c.2.2
d.none of these
Answer:
B. 22
Step-by-step explanation:
124.06 = 2.35h + 72.36
124.06 - 72.36 = 2.35h
51.7 = 2.35h
51.7/2.35 = h
22 = h
In a particular hospital, 5 newborn babies were delivered. here are their weights (in ounces): 119, 104, 92, 97, 103 Assuming these weights constitute an entire population find the standard deviation of the population, round answers to at least two decimal places.
Answer: standard deviation is 6.96
Step-by-step explanation:
Let m be mean
M=mean=sum/n
M=525/5
M=103
The standard deviation formula is :
S.D = sqrt( Summation of |x-m|^2 / n-1)
Let start finding:
|x-m|^2
For 1st: |119-103|^2=36
For 2nd: |104-103|^2=1
For 3rd: |92-103|^2=121
For 4th: |97-103|^2=36
For 5th: |103-103|^2=0
Summation of |x-m|^2 = 194
The standard deviation formula is :
S.D = sqrt( Summation of |x-m|^2 / n-1)
S.D= sqrt(194 / 4)
S.D=sqrt(48.5)
S.D= 6.96
The standard deviation is approximately 9.1 ounces.
Calculating the Population Standard Deviation
To find the standard deviation of the population of newborn babies' weights in a hospital, we need to follow several steps. The weights of the babies (in ounces) are 119, 104, 92, 97, and 103.
Calculate the mean (average) weight: (119 + 104 + 92 + 97 + 103) / 5 = 103
Determine the squared deviations from the mean for each weight: ((119 - 103)²), ((104 - 103)²), ((92 - 103)²), ((97 - 103)²), ((103 - 103)²).
Sum the squared deviations: 256 + 1 + 121 + 36 + 0 = 414.
Since we have the entire population, divide by the number of data points, which is 5: (414 / 5 = 82.8).
Take the square root of 82.8 to find the population standard deviation: √82.8} =approximately 9.1 ounces.
The population standard deviation of the babies' weights is approximately 9.1 ounces, rounded to two decimal places.
In each diagram below, determine whether the triangles are congruent, similar, but not congruent, or not similar. If you claim that the triangles are similar or congruent, make a flowchart justifying your answer.
Part a
Angle ABC = angle CDA (given by the angle markers)
Angle BAC = angle DCA (alternate interior angles)
Segment AC = segment AC (reflexive property)
Through AAS (angle angle side) we can prove the two triangles are congruent. We have a pair of congruent angles, and we have a pair of congruent sides that are not between the previously mentioned angles.
If two triangles are congruent, they are always similar as well (scale factor = 1).
The same cannot be said the other way around. Not all similar triangles are congruent.
Answer: Congruent======================================================
Part b
Angle FGH = angle JIH (both shown to be 50 degrees)
Angle FHG = angle JHI (vertical angles)
We have enough information to prove the triangles to be similar triangles. This is through the AA (angle angle) similarity rule. Since FG and JI are different lengths, this means the triangles are not congruent.
Answer: Similar but not congruent======================================================
Part c
For each right triangle shown, divide the longer leg over the shorter leg
larger triangle: (long leg)/(short leg) = 6/3 = 2
smaller triangle: (long leg)/(short leg) = 3/2 = 1.5
The two results are different, so the sides are not in proportion to one another, therefore the triangles are not similar.
Any triangles that are not similar will also never be congruent.
Answer: Not similar======================================================
Part d
Use the pythagorean theorem to find that PQ = 5 and KL = 12
We have two triangles with corresponding sides that are the same length
So we use the SSS (side side side) triangle congruence theorem to prove the triangles congruent. The triangles are also similar triangles (scale factor = 1)
Answer: Congruent======================================================
Summary of the answers:a. Congruentb. Similar but not congruentc. Not similard. CongruentIn Mathematics, triangles can be congruent, similar, or neither. Congruency means the triangles have the same three sides and angles. Similarity means the triangles have the same shape but not necessarily the same size.
Explanation:In Mathematics, particularly in Geometry, determining whether two triangles are congruent, similar, or neither is a pivotal concept. Triangles are congruent when they have exactly the same three sides and exactly the same three angles. On the other hand, triangles are similar when they have the same shape but not necessarily the same size.
To determine if triangles are congruent, you can use several postulates, including the Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA) postulates. For triangle similarity, the Angle-Angle (AA) postulate is often used. In the absence of sufficient information, the triangles cannot be declared similar or congruent.
A flowchart to justify the congruence or similarity would begin by assessing if all corresponding angles and sides match. If so, the triangles are congruent. If only the angles match and the sides are proportional, then the triangles are similar. In the absence of either, the triangles are neither similar nor congruent.
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A college has a 30% completion rate, meaning that 30% of all students who start at the college complete the goal they set. The president of the college sets a goal of increasing this number by 50%. What will the completion rate goal be as a percentage.
Answer:
45%
Step-by-step explanation:
For simplicity, let use assume there are 100 students in the school.
No. of students to complete college = (30/100) x 100 = 30 Students
President wants to increase by 50% = (50/100) x 30 = 15 Students
New set goal = 30 + 15 = 45 students.
Total number of students = 100 students
Therefore;
Rate goal % = (45/100) x 100% = 45%
Choco Dream is a firm that produces both dark chocolates as well as liquor chocolates. During a given month, the firm uses its resources to produce both varieties. Initially, the firm produced 5,000 bars of dark chocolates and 4,000 bars of liquor chocolates in a month. In order to increase production of the latter to 4,500, they had to reduce production of dark chocolates by 800 bars. When demand for liquor chocolates increased further, Choco Dream produced 5,000 bars of liquor chocolates and 3,200 bars of dark chocolates per month. Which of the following inferences can be drawn from the given information? A. Choco Dream's production possibilities frontier is linear. B. Both types of chocolates sold by Choco Dream are equally popular among consumers. C. Resources are equally productive in the production of both types of chocolates. D. The company is operating at one end of the PPF. E. Choco Dream faces increasing marginal opportunity cost in the production of liquor chocolates.
Answer:
E) Choco Dream faces increasing marginal opportunity cost in the production of liquor chocolates.
Step-by-step explanation:
When Choco Dream increased their production of liquor chocolates by 500 units (to 4,500 bars per month), their opportunity was 800 units of dark chocolate. But when they needed to increase liquor chocolates by 500 more units (to 5,000 bars per month), then the opportunity cost increased to 1,000 units of dark chocolate.
That means that for the first 500 extra liquor bars, the opportunity cost = 800 dark chocolate bars / 500 liquor bars = 1.6 dark chocolate bars for every extra liquor bar.
The second increased required a higher opportunity cost = 1,000 dark chocolate bars / 500 liquor bars = 2 dark chocolate bars for every extra liquor bar.
A swimming pool has to be drained for maintenance. The pool is shaped like a cylinder with a diameter of 5 m and a depth of 1.7 m. Suppose water is pumped out of the pool at a rate of 12 m3 per hour. If the pool starts completely full, how many hours will it take to empty the pool?
Answer:
2.78hrs
Step-by-step explanation:
Volume of water in the pool =πr2h
V = 3.142 * 2.5² *1.7
V = 33.38m³
Emptying the pool out at 12m³ per hour
= 33.38/12
= 2.78hrs
Let x1, x2, and x3 be 0 - 1 variables whose values indicate whether the projects are not done (0) or are done (1). Which answer below indicates that at least two of the projects must be done?a.x1+ x2+ x3>2b.x1+ x2+ x3<2c.x1+ x2+ x3= 2d.x1- x2= 0
Answer:
Correct statement: a. x₁ + x₂ + x₃ > 2
Step-by-step explanation:
The variables x₁, x₂ and x₃ takes value 0 if the projects are not done and 1 if the projects are done.
Consider that at least two projects are done, i.e. 2 or more projects are done.
This can happen in:
x₁ = 0, x₂ = 1 and x₃ = 1
x₁ = 1, x₂ = 0 and x₃ = 1
x₁ = 1, x₂ = 1 and x₃ = 0
x₁ = 1, x₂ = 1 and x₃ = 1
The statement (x₁ + x₂ + x₃ > 2) will be true only when all the variables takes the value 1.
This statement implies that 2 projects are definitely done.
Thus, the correct statement is (a).
Jeff sold the pumpkins he grew for $7 each at the farmers market. If Jeff sold 30 pumpkins how much money did he make. Write an expression to the amount of money in dollars Jeff made.
Jeff made $210 by selling 30 pumpkins.
Step-by-step explanation:
Given,
Selling price of each Pumpkin = $7
Number of pumpkins sold by Jeff = 30
We will multiply number of pumpkins sold by selling price per pumpkin.
Amount made by Jeff = Price per pumpkin * Number of pumpkins
Amount made by Jeff = 7 * 30 = $210
Therefore;
Jeff made $210 by selling 30 pumpkins.
Keywords: multiplication
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Peter takes16 minutes longer to mow the lawn by himself than Charles . Together they can mow the lawn in 18 minutes. How long will it take Charles to do it alone?
Answer:
about 29.7 minutes
Step-by-step explanation:
If it take c minutes for Charles to mow the lawn by himself, it takes c+16 minutes for Peter. The two of them working together can mow in one minute this fraction of the entire lawn:
1/c + 1/(c+16) = 1/18
Multiplying by 18c(c+16), we get ...
18(c +16) + 18(c) = c(c+16)/18
36c +288 = c^2 +16c
c^2 -20c = 288 . . . . . subtract 36c
c^2 -20c +100 = 388 . . . . . add (20/2)^2 = 100 to complete the square
(c -10)^2 = 388
c = 10 +√388 ≈ 29.6977 . . . . . take the positive square root
It takes Charles about 29.7 minutes to mow the lawn by himself.
Having trouble with this and 3 others
Answer:
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Step-by-step explanation:
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Solve for x. −6≥10−8x Enter your answer as an inequality in the box.
The solution of the expression of the inequality - 6 ≥ 10 - 8x for x
would be;
⇒ x ≥ 2
What is Mathematical expression?
The combination of numbers and variables by using operations addition, subtraction, multiplication and division is called Mathematical expression.
Given that;
The expression of the inequality is;
⇒ - 6 ≥ 10 - 8x
Now,
Solve the inequality for x as;
The inequality is;
⇒ - 6 ≥ 10 - 8x
Add 8x both side, we get;
⇒ - 6 + 8x ≥ 10 - 8x + 8x
⇒ - 6 + 8x ≥ 10
Add 6 both side, we get;
⇒ - 6 + 8x + 6 ≥ 10 + 6
⇒ 8x ≥ 16
Divide by 8 both side, we get;
⇒ x ≥ 16/8
⇒ x ≥ 2
Hence, - 6 ≥ 10 - 8x ⇒ x ≥ 2
Thus, The solution of the expression of the inequality - 6 ≥ 10 - 8x, for x will be;
⇒ x ≥ 2
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Which expression is equivalent to 30 (one-half x minus 2) + 40 (three-fourths y minus 4)? 45 x y minus 220 15 x minus 30 y minus 220 15 x + 30 y minus 220 15 x + 30 y minus 64
Answer:
The third option is correct i.e. 15 x + 30 y minus 220.
Step-by-step explanation:
We have to choose expression from the option that is equivalent to
[tex]30(\frac{1}{2}x - 2) + 40(\frac{3}{4}y - 4)[/tex]
Now, [tex]30(\frac{1}{2}x - 2) + 40(\frac{3}{4}y - 4)[/tex]
= 15x - 60 + 30y - 160
= 15x + 30y - 220
Therefore, the third option is correct i.e. 15 x + 30 y minus 220. (Answer)
Step-by-step explanation: C.
To rental car companies are running specials this month at Joseph rentals customers will pay $47 to rent a midsize car for the first day plus Two dollars for each additional day affair Fox rental the price for a midsize car is $36 for the first day and $13 for every additional day beyond that at some point renting from either one of these companies would cost to customer the same amount how many additional days would it take? How much with the customer pay?
Answer: it will take 2 days and the customer will pay $49
Step-by-step explanation:
Let x represent the number of days for which the cost would be the same.
At Joseph rentals, customers will pay $47 to rent a midsize car for the first day plus two dollars for each additional day. This means that the total cost of using Joseph rental for x days would be
47 + 2(x - 1) = 47 + 2x - 2
= 45 + 2x
At Fox rental, the price for a midsize car is $36 for the first day and $13 for every additional day beyond that. This means that the total cost of using Fox rental for x days would be
36 + 13(x - 1) = 36 + 13x - 13
= 23 + 13x
At the point where renting at either companies will cost the customer the same amount, then
45 + 2x = 23 + 13x
13x - 2x = 45 - 23
11x = 22
x = 22/11 = 2
The amount that the customer will psy is
23 + 13 × 2 = 49
Jamaica is considering buying either a car which goes 25 miles on 1 gallon of gas, or a truck that goes 10 miles on a gallon of gas. If gasoline costs $2.50 per gallon and Jamaica drives 1000 miles per month, how much less would it cost to drive the car each month than it would to drive the truck?
Answer: if he drives the car each month, he would spend $150 lesser than when he drives the truck.
Step-by-step explanation:
The car goes 25 miles on 1 gallon of gas. Jamaica drives 1000 miles per month, it means that the number of gallons of gas that he would use in a month is
1000/25 = 40 gallons of gas
If gasoline costs $2.50 per gallon and Jamaica chooses to buy a car, the cost of gas per month would be
2.5 × 40 = $100
The truck goes 10 miles on 1 gallon of gas. Jamaica drives 1000 miles per month, it means that the number of gallons of gas that he would use in a month is
1000/10 = 100 gallons of gas
If gasoline costs $2.50 per gallon and Jamaica chooses to buy a truck, the cost of gas per month would be
2.5 × 100 = $250
The difference between both costs is
250 - 100 = $150
Leon and Marisol biked the Brookside Trail to the end and back. Then they biked the Forest Glen Trail to the end and back before stopping to eat. How far did they bike before they stopped to eat?
The question is incomplete because you haven't attached the map with it. I am attaching the photo of the map here and answering according to it.
Answer:
Before stopping to eat, Leon and Marisol biked a total distance of 12 [tex]\frac{1}{3}[/tex] miles.
Step-by-step explanation:
Leon and Marisol first biked the Brookside Trail to the end an back. According to the map, this trail is 3 [tex]\frac{2}{3}[/tex] miles long and the distance from the trail to the end and back can be calculating by adding this distance twice. The distance is a mixed fraction and we need to convert it into a simple fraction to add it.
To convert the mixed fraction, we will first multiply the denominator with the whole number, i.e. 3x3 = 9 and then add the numerator to it i.e. 9 + 2 = 11. The new denominator will be the same as the previous denominator.
The fraction can now be written as [tex]\frac{11}{3}[/tex]. The distance of Brookside trail to the end and back is
[tex]\frac{11}{3}[/tex] +
Then, they biked the Forest Glen Trail to the end and back. The distance of this trail is 2 [tex]\frac{1}{2}[/tex] miles. We will add this distance twice as well to obtain the total distance traveled for this trail.
To convert the mixed fraction into a simple fraction, multiply the denominator with the whole number i.e. 2 x 2 = 4. Then add the numerator to this answer i.e. 4 + 1 = 5. This is the new numerator and the denominator stays the same. The fraction is [tex]\frac{5}{2}[/tex].
The distance of Forest Glen Trail to the end and back is:
[tex]\frac{5}{2}[/tex] +
The total distance traveled can be calculated by adding both the distances traveled in the individual trails. i.e.
[tex]\frac{22}{3}[/tex] + 5
This can be written as:
[tex]\frac{22}{3}[/tex] + [tex]\frac{5}{1}[/tex]
The denominators are different so we will find out the L.C.M (Lowest Common Multiple) of 3 and 1 which is 3.
We will multiply the numerator and denominator of second fraction with 3 to make the denominator equal to 3.
[tex]\frac{22}{3} + \frac{5 X 3}{1 X 3}[/tex]
= [tex]\frac{22}{3} + \frac{15}{3}[/tex]
= [tex]\frac{22+15}{3}[/tex]
= [tex]\frac{37}{3}[/tex]
To convert [tex]\frac{37}{3}[/tex] miles into a mixed fraction, divide 37 by 3 and write down the answer as a whole number, the remainder as the numerator and the previous denominator i.e. 3 as the new denominator.
3 x 12 = 36. So dividing 37 by 3 will yield 12 as the whole number. The remainder is 37-36 = 1. So, the mixed fraction will be 12 [tex]\frac{1}{3}[/tex]
Before stopping to eat, Leon and Marisol biked a total distance of 12 [tex]\frac{1}{3}[/tex] miles.
Answer:
Step-by-step explanation:
The question is incomplete because you haven't attached the map with it. I am attaching the photo of the map here and answering according to it.
Answer:
Before stopping to eat, Leon and Marisol biked a total distance of 12 miles.
Step-by-step explanation:
Leon and Marisol first biked the Brookside Trail to the end an back. According to the map, this trail is 3 miles long and the distance from the trail to the end and back can be calculating by adding this distance twice. The distance is a mixed fraction and we need to convert it into a simple fraction to add it.
To convert the mixed fraction, we will first multiply the denominator with the whole number, i.e. 3x3 = 9 and then add the numerator to it i.e. 9 + 2 = 11. The new denominator will be the same as the previous denominator.
The fraction can now be written as . The distance of Brookside trail to the end and back is
+
Then, they biked the Forest Glen Trail to the end and back. The distance of this trail is 2 miles. We will add this distance twice as well to obtain the total distance traveled for this trail.
To convert the mixed fraction into a simple fraction, multiply the denominator with the whole number i.e. 2 x 2 = 4. Then add the numerator to this answer i.e. 4 + 1 = 5. This is the new numerator and the denominator stays the same. The fraction is .
The distance of Forest Glen Trail to the end and back is:
+
The total distance traveled can be calculated by adding both the distances traveled in the individual trails. i.e.
+ 5
This can be written as:
+
The denominators are different so we will find out the L.C.M (Lowest Common Multiple) of 3 and 1 which is 3.
We will multiply the numerator and denominator of second fraction with 3 to make the denominator equal to 3.
=
=
=
To convert miles into a mixed fraction, divide 37 by 3 and write down the answer as a whole number, the remainder as the numerator and the previous denominator i.e. 3 as the new denominator.
3 x 12 = 36. So dividing 37 by 3 will yield 12 as the whole number. The remainder is 37-36 = 1. So, the mixed fraction will be 12
Before stopping to eat, Leon and Marisol biked a total distance of 12 miles.
Biologists tagged 103 fish in a lake january 1. On feburary 1, they returned and collected a random sample of 24 fish, 12 of which had been previously tagged. How many fish does the lake have
Answer:
206
Step-by-step explanation:
We have been given that Biologists tagged 103 fish in a lake January 1. On February 1, they returned and collected a random sample of 24 fish, 12 of which had been previously tagged.
To find the number of fish in the lake, we will use proportions because ratio of tagged fish and collected fish on February 1 will be equal to ratio of tagged fish and total fish on January 1.
[tex]\frac{\text{Tagged fish}}{\text{Collected fish}}=\frac{12}{24}[/tex]
Upon substituting the number of tagged fish in our proportion, we will get:
[tex]\frac{103}{\text{Total fish}}=\frac{12}{24}\\\\\frac{103}{\text{Total fish}}=\frac{1}{2}[/tex]
Cross multiply:
[tex]1\cdot \text{Total fish}=103\cdot 2\\\\\text{Total fish}=206[/tex]
Therefore, there are approximately 206 fishes in the lake.
The school store sells erasers. If Mrs. McBryde purchases 1000 erasers for $70.00, how much does 1 eraser cost? (Hint: Do not try to solve this with a long division problem!)
Answer:
The cost of 1 eraser is $0.07.
Step-by-step explanation:
Given:
The school store sells erasers. If Mrs. McBryde purchases 1000 erasers for $70.00,
Now, to find the cost of 1 eraser.
Let the cost of 1 eraser be [tex]x.[/tex]
The cost of 1000 erasers is $70.00.
So, 1000 is equivalent to $70.00.
Thus, 1 is equivalent to [tex]x.[/tex]
Now, to solve by using cross multiplication method:
[tex]\frac{1000}{70} =\frac{1}{x}[/tex]
By cross multiplying we get:
[tex]1000x=70[/tex]
Dividing both sides by 1000 we get:
[tex]x=\$0.07.[/tex]
Therefore, the cost of 1 eraser is $0.07.
Final answer:
The cost of one eraser is found by dividing the total cost of $70.00 by the number of erasers purchased, which is 1000, resulting in a cost of $0.07 per eraser.
Explanation:
To find the cost of one eraser, we divide the total cost by the number of erasers Mrs. McBryde purchased. She bought 1000 erasers for $70.00. So, we need to perform the division $70.00 ÷ 1000 erasers = $0.07 per eraser. This means each eraser costs 7 cents.
Alexa took out a $42,000 loan to remodel a house. The loan rate is 8.3% simple interest per year and will be repaid in six months. What is the maturity value that is paid back ?
Answer: The maturity value is $43743
Step-by-step explanation:
The formula for determining simple interest is expressed as
I = PRT/100
Where
I represents interest paid on the loan.
P represents the principal or amount that was taken as loan.
R represents interest rate.
T represents the duration of the loan in years.
From the information given,
P = 42000
R = 8.3
T = 6 months = 6/12 = 0.5 years
I = (42000 × 8.3 × 0.5)/100 = $1743
The maturity value is the total amount paid after the duration of the loan. It becomes
42000 + 1743 = $43743
please hurry
Which situations can represent the expression Check all that apply. Naomi gives some of her six pencils away. Sydney increased her collection of coins by six. Benjamin lost six of his stickers. Six servings of dinner were decreased by a number. Westville has 6 fewer schools than Eastville. Gabrielle decreased her 6-minute mile by an unknown amount of time.
Step-by-step explanation:
Hi,
Since there are multiple scenarios, lets first discuss the rules of developing expressions.
Any unknown value can be assumed as a variable.An increase means addition and decrease means subtraction.Using these rules we can develop the following expressions:
[tex]x - 6[/tex], where [tex]x[/tex] indicates the total number of pencils Naomi had.[tex]y + 6[/tex], where [tex]y[/tex] represents the number of coins Sydney had initially. [tex]z - 6[/tex], where [tex]z[/tex] refers to the total number of stickers Benjamin had.[tex]6 - a[/tex], where a is the number of servings decreased.[tex]b + 6,[/tex] where b is the number of schools in Eastville.[tex]6 - c[/tex], where c indicates the amount of time Gabrielle reduces.Tip:
In addition, the order of number doesn't matter however this is not the case in subtraction.