Answer:
(1) -0.833
(2) 0.80
(3) 0.70
(4) 390
(5) 90
(7) 48
Step-by-step explanation:
Given:
E (X) = 100, E (Y) = 120, E (Z) = 130
Var (X) = 9, Var (Y) = 16, Var (Z) = 25
Cov (X, Y) = -10, Cov (X, Z) = 12, Cov (Y, Z) = 14
The formulas used for correlation is:
[tex]Corr (A, B) = \frac{Cov (A, B)}{\sqrt{Var (A)\times Var(B)}} \\[/tex]
(1)
Compute the value of Corr (X, Y)-
[tex]Corr (X, Y) = \frac{Cov (X, Y)}{\sqrt{Var (X)\times Var(Y)}} \\=\frac{-10}{\sqrt{9\times16}} \\=-0.833[/tex]
(2)
Compute the value of Corr (X, Z)-
[tex]Corr (X, Z) = \frac{Cov (X, Z)}{\sqrt{Var (X)\times Var(Z)}} \\=\frac{12}{\sqrt{9\times25}} \\=0.80[/tex]
(3)
Compute the value of Corr (Y, Z)-
[tex]Corr (Y, Z) = \frac{Cov (Y, Z)}{\sqrt{Var (Y)\times Var(Z)}} \\=\frac{14}{\sqrt{16\times25}} \\=0.70[/tex]
(4)
Compute the value of E (3X+4Y-3Z)-
[tex]E(3X+4Y-3Z)=3E(X)+4E(Y)-3E(Z)\\=(3\times100)+(4\times120)-(3\times130)\\=390[/tex]
(5)
Compute the value of Var (3X-3Z)-
[tex]Var (3X-3Z)=[(3)^{2}\times Var(X)]+[(-3)^{2}\times Var (Z)]+(2\times3\times-3\times Cov(X, Z)]\\=(9\times9)+(9\times25)-(18\times12)\\=90[/tex]
(6)
Compute the value of Var (3X+4Y-3Z)-
[tex]Var (3X+4Y-3Z)=[(3)^{2}\times Var(X)]+[(4)^{2}\times Var(Y)]+[(-3)^{2}\times Var (Z)]+[(2\times3\times4\times Cov(X, Y)]+[(2\times3\times-3\times Cov(X, Z)]+[(2\times4\times-3\times Cov(Y, Z)]\\=(9\times9)+(16\times16)+(9\times25)+(24\times-10)-(18\times12)-(24\times14)\\=-230[/tex]
But this is not possible as variance is a square of terms.
(7)
Compute the value of Cov (3X, 2Y+3Z)-
[tex]Cov(3X, 2Y+3Z)=Cov(3X,2Y)+Cov(3X, 3Z)\\=6Cov(X, Y)+9Cov(X,Z)\\=(6\times-10)+(9\times12)\\=48[/tex]
The correct answers to the given set of data are:
(1) -0.8332) 0.80(3) 0.70(4) 390(5) 90(7) 48What is Variance?This refers to the measurement of spread between numbers which can be found in a set of data.
Hence, to compute the variance and covariance
E (X) = 100, E (Y) = 120, E (Z) = 130Var (X) = 9, Var (Y) = 16, Var (Z) = 25Cov (X, Y) = -10, Cov (X, Z) = 12, Cov (Y, Z) = 14Using the variance formula we can see that the given sets of data are:
-0.8330.800.703909048 respectivelyRead more about variance here:
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A square park has a diagonal walkway from one corner to another. If the walkway is 120 meters long, what is the approximate length of each side of the park?
Answer:
85 m
Step-by-step explanation:
The diagonal of a square is √2 times the length of the side. The park will have a side length of 120/√2 m ≈ 84.85 m, about 85 meters.
_____
The relations are ...
diagonal = (√2)×(side length)
side length = diagonal/√2 . . . . . . . . . divide the above equation by √2
Use well-ordering property to prove that if n people stand in a line, where n is a positive integer, and if the first person in the line is a woman and the last person in line is a man, then somewhere in the line there is a woman directly in front of a man.
Answer:
The people forming the line form a well ordered set, this means, every non empty subset has a minimum. In other words, in any smaller (but non empty) set of people there will always be one that is last in the line.
Lets take the subset formed of all the Women in the line. This subset in non empty because there is a women in the line (the first person in the line is a women), therefore, there is a women that is behind every other women in the line; this women can only have men behind, if any. If she had no men behind her, then she should be last in the line, bacause the doesnt have any men or women behind her. This is a contradiction because there is a man at the end of the line, not a woman. This shows that this woman that is behind every other woman has men behind, in particularly immediately behind, her.
Using well - ordering property in mathematical induction, we have been able to prove that; if n people stand in a line, where n is a positive integer, then somewhere in the line there is a woman directly in front of a man.
Mathematical inductionMathematical induction is defined as a proof technique that is used to prove that a property Pₙ will hold for every natural number such as n = 0, 1, 2, 3 e.t.c.
Now, we want to prove that If n people stand in a line, where n is a positive integer, and if the first person in the line is a woman and the last person in line is a man, then somewhere in the line, there is a woman directly in front of a man.
Now, let us deduce as follows;
When n = 2; This means that the first person is a woman and last is man.When n = k (k ≥ 2); This means there is a woman in front of a manThus;
When n = k + 1, what we will get is that;
- A new person is before a woman that is directly in front of a man.
- The new person is after the man directly after the woman.
- A new person is added between the woman and the man.
In conclusion;
For every positive integer n(n > 2), there is always a woman directly in front of a man.
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what is the solution to the following expression if x=5
The value of given expression is 16
Solution:
Given expression is:
[tex]\sqrt{x^2} + \sqrt{x} \times \sqrt{x} + 6[/tex]
We have to find the value of expression when x = 5
Substitute x = 5
[tex]\sqrt{x^2} + \sqrt{x} \times \sqrt{x} + 6\\\\\sqrt{5^2} + (\sqrt{5} \times \sqrt{5}) + 6[/tex]
We know that,
[tex]\sqrt{a} \times \sqrt{a} = a[/tex]
Therefore, the above equation becomes,
[tex]\sqrt{5^2} + 5 + 6\\\\\sqrt{25} + 5 + 6\\\\5 + 5 + 6 = 10 + 6 = 16[/tex]
Thus the value of given expression is 16
Tamira has $55 in her lunch account. Each day she spends $5 on lunch. Which equation represents the amount of money, y, Tamira has left in her lunch account after x days?
Answer:
y=-5x+55
Step-by-step explanation:
i had a quiz and i got the correct answer on the test corrections your welcome
One container is filled with a mixture that is 30%acid a second container is filled with a mixture that is 50%acid the second containet is 50%larger than the first and that twocontainers are are emptied into a third what percent of acid is the third container
Answer:
The amount of acid in third container is =42%
Step-by-step explanation:
Given , one container is filled with a mixture that is 30% acid a second container filled with a mixture that is 50% acid and the second container 50% larger than the first .
Let, the volume of first container is = x
Then , the volume of second container = (x+ x of 50%)
= x + 0.5 x
= 1.5 x
Therefore the amount of acid in first container [tex]=x \times \frac{30}{100}[/tex] = 0.3 x
The amount of acid in second container [tex]=1.5x \times \frac{50}{100}[/tex] = 0.75x
Total amount of acid= 0.3x + 0.75x = 1.05 x
Total amount of solution = x+1.5x = 2.5x
The amount of acid in third container is = [tex]\frac{1.05x}{2.5x} \times 100%[/tex] %
= 42%
Final answer:
By calculating the total acid content and total volume when two containers with different acid concentrations are mixed, we find that the third container has an acid concentration of 42%.
Explanation:
The question involves mixing two solutions of different acid concentrations and calculating the final concentration of acid. Let's assume the first container has 1 unit of volume. Since the second container is 50% larger, it has 1.5 units of volume. The first container has 30% acid, so it contains 0.3 units of acid. The second container has 50% acid, resulting in 0.75 units of acid. Together, the total volume of the solution is 2.5 units, and the total amount of acid is 1.05 units.
Therefore, the percentage of acid in the third container is calculated by dividing the total acid by the total volume: (1.05 units of acid / 2.5 units of volume) x 100% = 42%.
Write a formula that describes the value of an initial investment of $4,000 that loses value at a rate of 10% per year, compounded continuously.
Answer:
see below
Step-by-step explanation:
The formula is the same whether the rate of change is positive or negative. Here, it is negative.
A = Pe^(rt)
for P = 4000, r = -0.10. Then you have ...
A = 4000e^(-0.10t)
In a certain corporation, 2/3 of the employees are men and 1/3 are women. Of the men, 5/8 are college-educated and 3/8 are not. Of the women, 2/5 are college-educated and 3/5 are not. What percentage of the employees are college-educated? What percentage of the college-educated employees are women?
Answer:
55% and 24%
Step-by-step explanation:
Firstly, we need an arbitrary number to serve as the number of students.
Let the total number of students be 120.
Firstly we need college-educated percentage.
Men are two-thirds, women are one-third
Men = 2/3 * 120 = 80
Women = 1/3 * 120 = 40
5/8 of men are college educated = 5/8 * 80 = 50
2/5 of women are college educated = 2/5 * 40 = 16
Total college educated = 50+ 16 = 66
% college educated = 66/120 * 100 = 55%
Percentage of college educated that are women.
Total college educated = 66
The % is 16/66 * 100 = 24%
There are 40 math teachers, 45 English teachers, 27 science teachers, and 32 social studies teachers. The superintendent of the region wants to select a board of 20 teachers. How many science teachers should he select?
Answer:
Superintendent of the region will select approximately 4 Science teachers in the board.
Step-by-step explanation:
Given:
Number of math teachers = 40
Number of English teachers = 45
Number of Science teachers = 27
Number Social Studies teachers = 32
Number of teachers to be selected in the board =20
We need to find the number Science teachers should be selected in the board.
Solution:
First we ill find the Total number of teachers.
Now we can say that;
Total number of teachers is equal to sum of Number of math teachers, Number of English teachers,Number of Science teachers and Number Social Studies teachers
framing in equation form we get;
Total number of teachers = [tex]40+45+27+32=144[/tex]
Probability of Science teacher can be calculated by Dividing Number of Science teacher by Total number of teachers.
[tex]P(S)=\frac{27}{144}=0.1875[/tex]
Now the number of Science teachers should be selected in the board we need to multiply P(S) with Number of teachers to be selected in the board.
framing in equation form we get;
number of Science teachers should be selected in the board = [tex]0.1875\times 20=3.75\approx4[/tex]
Hence We can say that;
Superintendent of the region will select approximately 4 Science teachers in the board.
Calculate the proportion of science teachers out of the total and apply it to the board size to determine the number of science teachers to select.
To determine how many science teachers the superintendent should select for a board of 20 teachers:
Calculate the total number of teachers: 40 math + 45 English + 27 science + 32 social studies = 144 teachersFind the proportion of science teachers: 27 science teachers out of 144 total teachers = 0.1875Apply the proportion to the board size: 0.1875 * 20 = 3.75, so the superintendent should select 4 science teachers.PLEASE HELP QUICKLY! 20 points! I will mark best answer as Brainliest!
A pitcher throws a baseball straight into the air with a velocity of 72 feet/sec. If acceleration due to gravity is -32 ft/sec^2, how many seconds after it leaves the pitcher's hand will it take the ball to reach its highest point? Assume the position at time t = 0 is 0 feet.
t= 2.25 secs
Step-by-step explanation:
Step 1 :
Equation for motion with uniform acceleration is v = u+ at
where v is the final velocity
u is the initial velocity
a is the acceleration due to gravity
and t is the time
Step 2 :
Here , v = 0 because at the highest point final velocity is 0.
u = 72 feet/sec
a = -32 ft/sec^2
We need to find the time t.
Substituting in the equation we have,
0 = 72 -32 * t
=> 32 t = 72
=> t = 72/32 = 2.25 secs
According to the U.S. Census bureau, 23.5% of people in the United States are under the age of 18. In a random sample of 250 residents of a small town in Ohio, 28% of the sample was under 18. Which one of the following statements is true?
1. 23.5% and 28% are statistics, 250 and 18 are parameters
2. 23.5% and 28% are parameters, 250 and 18 are statistics
3. 23.5% and 28% are parameters, 18 is a statistic
4. 28% is a parameter and 23.5% is a statistic
5. 23.5% is a parameter and 28% is a statistic
Answer:
Option 5) 23.5% is a parameter and 28% is a statistic
Step-by-step explanation:
We are given the following in the question:
"According to the U.S. Census bureau, 23.5% of people in the United States are under the age of 18. In a random sample of 250 residents of a small town in Ohio, 28% of the sample was under 18."
Population is the collection of all the observation of individual of interest.Population of interest:
People in the United States
Parameter is any value that describes the population.Parameter:
23.5% of people in the United States are under the age of 18
Sample is a subset of a population and is always smaller than the population.Sample:
Random sample of 250 residents of a small town in Ohio
Statistic is any value that describes a sample.Statistic:
28% of the sample was under 18
Thus, the correct answer is
Option 5) 23.5% is a parameter and 28% is a statistic
The residents of a downtown neighborhood designed a triangular-shaped park as part of a city beautification program. The park is bound by streets on all sides. The second angle of the triangle is 7° more than the first. The third angle is 7° less than seven times the first. Find the measures of the angles.
The measure of the first, second and third angles are 20°, 27° and 133°.
What is an exterior angle of a triangle ?An exterior angle of a triangle is the sum of two opposite interior angles.
According to given question
The residents of a downtown neighbourhood designed a triangular-shaped park as part of a city beautification program.
Let us assume the first angle to be x°.
Therefore from the given data second angle is (x + 7)° and the third angle is (7x - 7)°.
We know that the sum of all the interior angles of a given triangle is 180°.
∴ x + (x + 7) + (7x - 7) = 180°
9x = 180°
x = 180°/9
x = 20°.
So, The first angle of the triangular park is 20°, The second angle is 27° and the third angle is 133°.
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If θ is an angle in standard position that terminates in Quadrant IV such that cosθ = 3/5, then cosθ/2 = _____. -
Answer:
[tex]cos \frac{\theta}{2}= 0.894[/tex].
Step-by-step explanation:
Given:
[tex]cos\ \theta= \frac35[/tex]
We need to find [tex]cos \ \frac{\theta}{2}[/tex]
Solution:
[tex]cos\ \theta= \frac35[/tex]
First we will find the value of [tex]\theta[/tex].
Now taking [tex]cos^{-1}[/tex] on both side we get;
[tex]\theta= cos^{-1} \ \frac{3}{5}\\\\\theta = 53.13[/tex]
Now we will find the [tex]\frac{\theta}{2}[/tex].
[tex]\frac{\theta}{2}[/tex] = [tex]\frac{53.13}{2} = 26.565[/tex]
Now we will find [tex]cos \ \frac{\theta}{2}[/tex] we get;
[tex]cos \frac{\theta}{2}=cos\ 26.565 = 0.894[/tex]
Hence [tex]cos \frac{\theta}{2}= 0.894[/tex].
Answer:
+/- (2\sqrt(5))/(5)
Step-by-step explanation:
The radius of a cylindrical gift box is (4x + 1) inches. The height of the gift box is three times the radius. What is the surface area of the cylinder? Write your answer as a polynomial in standard form.
Answer:
S = 128πx² + 64πx + 8π
Step-by-step explanation:
Suraface area of a cylinder is given by:
S = 2πrh + 2πr²
We know that the height is 3 times as big as the radius, hence:
h = 3r
so we can plug in the new h value and rewrite the S equation as:
S = 2πrh + 2πr²
S = 2πr(3r) + 2πr²
S = 6πr² + 2πr²
S = 8πr²
We're given in the question that the radius is (4x + 1) inches, so plug that into r.
Given: r = 4x + 1
Therefore,
S = 8πr²
S = 8π(4x + 1)²
S = 8π(16x²+8x+1)
S = 128πx² + 64πx + 8π
Answer:
The answer to your question is 128πx² + 64πx + 8π
Step-by-step explanation:
Data
radius = (4x + 1)
height = 3(4x + 1)
Formula
Area = 2πrh + 2πr²
Substitution
Area = 2π(4x + 1)(3)(4x + 1) + 2π(4x + 1)
Simplification
Area = 6π(4x + 1)² + 2π(4x + 1)²
Area = 6π(16x² + 8x + 1) + 2π(16x² + 8x + 1)²
Area = 96πx² + 48πx + 6π + 32πx² + 16πx + 2π
Area = 128πx² + 64πx + 8π
How do you do this question?
Answer:
E) 13
Step-by-step explanation:
∫₀⁴ f'(t) dt = f(4) − f(0)
8 = f(4) − 5
f(4) = 13
A survey was conducted by a website on which online music channels subscribers use on a regular basis. The following
information summarizes the answers.
10 listened to rap, heavy metal, and alternative rock.
13 listened to rap and heavy metal.
16 listened to heavy metal and alternative rock.
26 listened to rap and alternative rock.
37 listened to rap.
30 listened to heavy metal.
49 listened to alternative rock.
18 listened to none of these three channels.
a. How many people were surveyed?
b. How many people listened to either rap or alternative rock?
c. How many listened to heavy metal only?
The total number of people surveyed is 134. The number of people who listened to either rap or alternative rock is 76. The number of people who listened to heavy metal only is 1.
Explanation:To solve this problem, we can use a Venn diagram to organize the given information. Let's start by filling in the total number of people surveyed, which is the sum of those who listened to rap, heavy metal, alternative rock, and none of the three channels. From the given information, we have:
Rap: 37 Heavy Metal: 30 Alternative Rock: 49 None: 18
To find the total number of people surveyed, we add these numbers:
37 + 30 + 49 + 18 = 134.
Therefore, 134 people were surveyed.
b. To find the number of people who listened to either rap or alternative rock, we need to add the number of people who listened to rap and the number of people who listened to alternative rock, while making sure we don't double count those who listened to both:
37 + 49 - 10 = 76.
Therefore, 76 people listened to either rap or alternative rock.
c. To find the number of people who listened to heavy metal only, we need to subtract those who listened to heavy metal and either rap or alternative rock from the total number of people who listened to heavy metal:
30 - 13 - 16 = 1.
Therefore, 1 person listened to heavy metal only.
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A line perpendicular to another line or to a tangent line is called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curve at the given point P.
Y=x^2-3x
P (7, 28)
The equation of the normal line at P(7, 28) is:
Answer:
The equation of the normal line to the curve y = x² - 3x at the point (7, 28) is 11y + x - 315 = 0
Step-by-step explanation:
First, we need to find the slope of the line tangent to the curve at the point (7, 28).
To do this, we need to differentiate y with respect to x and evaluate at the point x0 = 7.
Given y = x² - 3x
dy/dx = 2x - 3
So, the slope, m of the tangent line is dy/dx at x = 7:
m = 2(7) - 3 = 11
Slope, m = 11
The slope, m2 of the line perpendicular to the tangent is given as m2 = -1/m
m2 = −1/11
Finally, given the slope m of a line, and a point (x0, y0) on the line, we can use the point-slope form of the equation of a line:
y - y0=m(x - x0)
The perpendicular line has slope m2 = -1/11, and (7, 28) is a point on that line, the desired equation is:
y - 28 = (-1/11)(x - 7)
Or multiplying by 11, we have
11y - 308 = -x + 7
11y + x - 315 = 0
The slope of the curve at point (7,28) is 11, found by differentiating the given function. The slope of the normal line is the negative reciprocal of that, -1/11. Inserting these into the line equation, we get y - 28 = -1/11 * (x - 7).
Explanation:To find the equation of the line perpendicular to the tangent line, we first find the slope of the tangent line. The derivative of the given function y = x^2 - 3x tells us the slope of the curve at any point, so let's find the derivative:
y' = 2x - 3
Now, let's find the slope of the tangent line at the point (7,28) by plugging 7 into the derivative:
y'(7) = 2*7 - 3 = 11
The slope of the line perpendicular to this (the normal line) is the negative reciprocal of the tangent line's slope, so it's -1/11.
The equation of the line with slope m that goes through the point (x1, y1) is:
y - y1 = m*(x - x1)
Substitute the slope of -1/11 and the point (7,28) into this equation to get the equation of the normal line:
y - 28 = -1/11 * (x - 7)
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Delegates from 10 countries, including Russia,
France, England, and the United States, are to
be seated in a row. How many different seating arrangements are possible if the French and
English delegates are to be seated next to each
other and the Russian and U.S. delegates are not
to be next to each other
Answer:
564,480
Step-by-step explanation:
The appliocation of factorial n! = n(n-1)(n-2)(n-3).....
Arrangement when french and English delegates are to seat together ;
since we have 10 delegates, Freench and english will be treated as 1 delegates as such we have 9groups. The number of ways of arranging 9groups = 9! and if the french and English are to be treated as a group = 2!
Hence, number of ways of arrangement = 9! x 2! = 725,760
In this case, English and french are to be seated next to each other while russia and US delegates are not to seat nect to each other ; both groups will be treated as 2 as such we have 8groups, the number of ways of arranging 8groups = 8!
french and english together = 2!
Russia and US together = 2!
The number of ways of arrangement = 8! x 2! x 2! = 161,280
hence the number of ways of arranging 10 delegates if french and english are to be seated next to each other and Russia and US are not to be seated next to each other;
N = 725,760 - 161,280 = 564,480
There are 282,240 different seating arrangements possible if the French and English delegates are to be seated next to each other and the Russian and U.S. delegates are not to be next to each other.
Explanation:To determine the number of different seating arrangements, we can treat the French and English delegates as a single entity. Therefore, we have 9 entities overall (Russia, (France and England), United States, and the 6 remaining countries).
The number of arrangements of these 9 entities is 9! (9 factorial) which is equal to 362,880. However, since the Russian and U.S. delegates cannot be seated next to each other, we need to subtract the number of arrangements where they are together.
Let's consider the Russian and U.S. delegates as a single entity as well. Now we have 8 entities to arrange (Russian and U.S., (France and England), and the 6 remaining countries). The number of arrangements of these 8 entities is 8! which is equal to 40,320.
Finally, we need to consider the arrangements of the Russian and U.S. delegates within their entity. The Russian and U.S. delegates can be arranged in 2! (2 factorial) ways, which is equal to 2.
Therefore, the total number of different seating arrangements is 362,880 - (40,320 * 2) = 362,880 - 80,640 = 282,240.
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Mr.Nguyen bought a suit that was on sale for 40% off the original price.He paid 9% sales tax on the sale price. The original price of the suit was 260$ how much did he pay for the suit, including tax, to the nearest dollar?
Answer: The total amount that he paid for the suit is $170
Step-by-step explanation:
The original price of the suit was $260.
Mr.Nguyen bought a suit that was on sale for 40% off the original price. This means that the amount of discount in the suit is
40/100 × 260 = 0.4 × 260 = $104
The sale price would be
260 - 104 = $156
He paid 9% sales tax on the sale price. This means that the amount of sales tax that he paid is
9/100 × 156 = 0.09 × 156 = $14.04
The total amount that he paid for the suit, including tax is
156 + 14.04 = $170 to the nearest dollar
The drawing plan for an art studio shows a rectangle that is 19.2 inches by 6 inches. The scale in the plan is 3 in.: 5 ft. Find the length and width of the actual studio. Then find the area of the actual studio.
Answer:
321.24 ft²
Step-by-step explanation:
If 3in = 5ft
1in = x
5/3 = 1.6
So 1in = 1.67 ft
Actual dimensions will be;
L = 19.2*1.67 = 32.06ft
W = 6*1.67 = 10.02ft
Area = 32.06 * 10.02 = 321.24 ft²
To find the length and width of the actual studio, set up a proportion to convert the given dimensions from the scale to the actual dimensions. Then, find the area of the actual studio by multiplying the length and width.
Explanation:To find the length and width of the actual studio, we need to convert the given dimensions from the scale to the actual dimensions. Since the scale is 3 in.: 5 ft, we can set up the proportion:
3 in / 5 ft = 19.2 in / x ft
Cross multiplying and solving for x, we get:
x = 32 ft
The length of the actual studio is 32 ft. Similarly, we can find the width:
3 in / 5 ft = 6 in / y ft
Solving for y, we get:
y = 10 ft
The width of the actual studio is 10 ft.
To find the area of the actual studio, we multiply the length and width:
Area = length x width = 32 ft x 10 ft = 320 ft²
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The surface area of a cube is increasing at a rate of 15 square meters per hour. At a certain instant, the surface area is 24 square meters. What is the rate of change of the volume of the cube at that instant (in cubic meters per hour)?
Answer:
The volume of cube is increasing at a rate 7.5 cubic meter per hour.
Step-by-step explanation:
We are given the following in the question:
Surface area of cube = 24 square meters.
Let l be the edge of cube.
Surface area of cube =
[tex]6l^2 = 24\\l^2 = 4\\l = 2[/tex]
Thus, at that instant the edge of cube is 2 meters.
[tex]\dfrac{dS}{dt} = 15\text{ square meters per hour}\\\\\dfrac{d(6l^2)}{dt} = 15\\\\12l\dfrac{dl}{dt} = 15\\\\\dfrac{dl}{dt} = \dfrac{15}{12\times 2} = \dfrac{15}{24}[/tex]
We have to find the rate of change in volume.
Volume of cube =
[tex]l^3[/tex]
Rate of change of volume =
[tex]\dfrac{dV}{dt} = \dfrac{d(l^3)}{dt} = 3l^2\dfrac{dl}{dt}\\\\\dfrac{dV}{dt} = 3(2)^2\times \dfrac{15}{24} \\\\\dfrac{dV}{dt} = 7.5\text{ cubic meter per hour}[/tex]
Thus, the volume of cube is increasing at a rate 7.5 cubic meter per hour.
The rate of change of the volume of the cube at the given instant is; dV/dt = 7.5 m³/h
Surface area is increasing at a rate of 15 m²/h
Thus, dS/dt = 15 m²/h
Now, at a certain instant the surface area of the cube is 24 m².
Formula for cube surface area is; S = 6l²
Thus;
6l² = 24
l² = 24/6
l² = 4
l = √4
l = 2 m
From S = 6l², differentiating both sides with respect to t gives;
dS/dt = 12l(dl/dt)
We know that dS/dt = 15
Thus;
12l(dl/dt) = 15
dl/dt = 15/(12l)
putting 2 for l gives;
dl/dt = 15/(12 * 2)
dl/dt = 0.625 m/h
Now formula for volume of a cube is; V = l³
Differentiating both sides with respect to t gives;
dV/dt = 3l²(dl/dt)
plugging in then relevant values gives;
dV/dt = 3 × 2² × (0.625)
dV/dt = 7.5 m³/h
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The graph below shows the height of a tunnel f(x), in feet, depending on the distance from one side of the tunnel x, in feet:
Graph of quadratic function f of x having x-intercepts at ordered pairs 0, 0 and 36, 0. The vertex is at 18, 32.
Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height? (6 points)
Part B: What is an approximate average rate of change of the graph from x = 5 to x = 15, and what does this rate represent? (4 points)
Answer:
Part A) see the explanation
Part B) see the explanation
Step-by-step explanation:
Let
x ----> the distance from one side of the tunnel in feet:
f(x) ---> the height of a tunnel in feet
Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the distance and height?
1)
we know that
The x-intercepts are the values of x when the value of the function is equal to zero
In the context of the problem, the x-intercepts are the distances from one side of the tunnel when the height of the tunnel is equal to zero
2)
The maximum value of the graph is the vertex
The x-coordinate of the vertex represent the distance from one side of the tunnel when the height is a maximum value
The y-coordinate of the vertex represent the maximum height of the tunnel
In this problem
the vertex is (18,32)
That means
The maximum height of the tunnel is 32 feet and occurs when the distance from one side of the tunnel is 18 feet
3)
we know that
The function is increasing at the interval [0,18)
That means
As the distance from one side of tunnel increases the height of the tunnel increases too
The function is decreasing at the interval (18,36]
That means
As the distance from one side of tunnel increases the height of the tunnel decreases
Part B: What is an approximate average rate of change of the graph from x = 5 to x = 15, and what does this rate represent?
we know that
To find the average rate of change, we divide the change in the output value by the change in the input value
the average rate of change using the graph is equal to
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
In this problem we have
[tex]a=5[/tex]
[tex]b=15[/tex]
[tex]f(a)=f(5)=15[/tex] ----> see the attached figure
[tex]f(b)=f(15)=31[/tex] ----> see the attached figure
Substitute
[tex]\frac{31-15}{15-5}=1.6[/tex]
That means
The height of the tunnel increases 1.6 feet as the distance from one side of tunnel increases 1 foot in the interval from x=5 to x=15
A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. A specific design is randomly generated by the Web server when you visit the site. Let A denote the event that the design color is red, and let B denote the event that the font size is not the smallest one. Use the addition rules to calculate the following probabilities.
A. P(A ∪ B).
B. P(A ∪ B').
C. P(A' ∪ B').
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.
The list shows the number of viewers of an online music video each day for 5 consecutive days. 5; 35; 245; 1,715; 12,005; By what factor did the number of viewers change each day to the first day to the fifth day?
Answer: 7
Step-by-step explanation:
Given : The list shows the number of viewers of an online music video each day for 5 consecutive days.
5; 35; 245; 1,715; 12,005;
In exponential growth equation : [tex]y=Ab^x[/tex] , A = initial value , b= growth factor and x = time period.
In the given table A = 5 , at x = 0
Growth factor is the common ratio between the consecutive terms.
So [tex]b=\dfrac{35}{5}=7[/tex]
and we can check that it is common between each consecutive terms is 7.
Hence, the factor by which the number of viewers change each day to the first day to the fifth day = 7
The number of viewers changed by a factor of 2401 from the first day to the fifth day.
Explanation:The number of viewers of an online music video changed each day by multiplying the previous day's viewers by a constant factor. To find this factor, we divide the viewers on the fifth day by the viewers on the first day. By doing this, we get a factor of 12005/5 = 2401.
So, the number of viewers changed by a factor of 2401 from the first day to the fifth day.
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The range for a list of measurements equals the greatest measurement minus the least measurement. The two lists of measurements, X and Y, combined consist of 30 measurements with a range of 25. What is the range for the measurements in list X?
Answer:
Insufficient information
Step-by-step explanation:
The information provided us isn't sufficient to determine the range for the measurements in list X. Because we need to be given the range for the measurements in Y at least. We also need the number of measurements in both X and Y.
Select all expressions that represent a correct solution to the equation 6(x+4)=20. copied for free from openupresources.Org Select all that apply: A. (20−4)÷6 B. 16(20−4) C. 20−6−4 D. 20÷6−4 E. 1/6(20−24) F. (20−24)÷6
Option D: [tex]20 \div 6-4[/tex] is the correct solution to the equation [tex]6(x+4)=20[/tex]
Option E: [tex]\frac{1}{6} (20-24)[/tex] is the correct solution to the equation [tex]6(x+4)=20[/tex]
Option F: [tex](20-24) \div 6[/tex] is the correct solution to the equation [tex]6(x+4)=20[/tex]
Explanation:
The expression is [tex]6(x+4)=20[/tex]
Let us find the value of x.
[tex]\begin{aligned}6(x+4) &=20 \\6 x+24 &=20 \\6 x &=-4 \\x &=-\frac{2}{3}\end{aligned}[/tex]
Now, we shall find the expression that is equivalent to the value [tex]x=-\frac{2}{3}[/tex]
Option A: [tex](20-4) \div 6[/tex]
Simplifying the expression, we have,
[tex]\frac{16}{6}=\frac{8}{3}[/tex]
Since, [tex]\frac{8}{3}[/tex] is not equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex](20-4) \div 6[/tex] is not equivalent to the equation [tex]6(x+4)=20[/tex]
Hence, Option A is not the correct answer.
Option B: [tex]16(20-4)[/tex]
Simplifying the expression, we have,
[tex]16(16)=256[/tex]
Since, 256 is not equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex]16(20-4)[/tex] is not equivalent to the equation [tex]6(x+4)=20[/tex]
Hence, Option B is not the correct answer.
Option C: [tex]20-6-4[/tex]
Simplifying the expression, we have,
[tex]20-10=10[/tex]
Since, 10 is not equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex]20-6-4[/tex] is not equivalent to the equation [tex]6(x+4)=20[/tex]
Hence, Option C is not the correct answer.
Option D: [tex]20 \div 6-4[/tex]
Using PEMDAS and simplifying the expression, we have,
[tex]$\begin{aligned}(20 \div 6)-4 &=\frac{10}{3}-4 \\ &=\frac{10-12}{3} \\ &=-\frac{2}{3} \end{aligned}$[/tex]
Thus, [tex]-\frac{2}{3}[/tex] is equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex]20 \div 6-4[/tex] is equivalent to the equation [tex]6(x+4)=20[/tex]
Hence, Option D is the correct answer.
Option E: [tex]\frac{1}{6} (20-24)[/tex]
Simplifying the expression, we have,
[tex]$\begin{aligned} \frac{1}{6}(20-24) &=\frac{1}{6}(-4) \\ &=-\frac{4}{6} \\ &=-\frac{2}{3} \end{aligned}$[/tex]
Thus, [tex]-\frac{2}{3}[/tex] is equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex]\frac{1}{6} (20-24)[/tex] is equivalent to the equation [tex]6(x+4)=20[/tex]
Hence, Option E is the correct answer.
Option F: [tex](20-24) \div 6[/tex]
Simplifying the expression, we have,
[tex]$\begin{aligned}(20-24) \div 6 &=-\frac{4}{6} \\ &=-\frac{2}{3} \end{aligned}$[/tex]
Thus, [tex]-\frac{2}{3}[/tex] is equivalent to [tex]x=-\frac{2}{3}[/tex], the expression [tex](20-24) \div 6[/tex] is equivalent to the equation [tex]6(x+4)=20[/tex]
Hence, Option F is the correct answer.
To solve 6(x + 4) = 20, we isolate x to get x = -2/3. The correct expressions representing the solution are E. 1/6(20 - 24) and F. (20 - 24) / 6.
To solve the equation 6(x + 4) = 20, we need to isolate x.
First, distribute the 6 on the left-hand side:
6x + 24 = 20
Next, subtract 24 from both sides:
6x = -4
Finally, divide by 6:
x = -4/6 = -2/3
Now, let's analyze the given choices:
A. (20 - 4) ÷ 6: This simplifies to 16 / 6, which does not match x = -2/3.B. 16(20 - 4): This simplifies to 16 × 16, which is incorrect.C. 20 - 6 - 4: This simplifies to 10, which is incorrect.D. 20 ÷ 6 - 4: This simplifies to 10/3 - 4, which is incorrect.E. 1/6(20 - 24): This simplifies to 1/6(-4) = -2/3, which matches the solution.F. (20 - 24) / 6: This simplifies to -4/6 = -2/3, which matches the solution.Therefore, the correct choices are E and F.
Sally and Adam work at different jobs. Sally earns $5 per hour and Adam earns $4 per hour. They each earn the same amount per week but Adam works 2 more hours. How many hours a week does Adam work?
Answer:
10 hrs
Step-by-step explanation:
sally earns $5/hr while Adam earns $4/hr
let the number of Hour sally works be 'x'
From question Adam work 2 hrs more than sally,
therefore Adam works (x + 2)hrs also the both earn same amount per week
therefore;
4 x [tex](x + 2)[/tex] = 5 x [tex]x[/tex]
4x + 8 = 5x
5x - 4x = 8
x = 8hrs
Adam works for (8+2)hrs = 10hrs
Find the volume of a cube with side length of 7 in.
A: 147 in³ B: 49 in³
C:215 in³ D: 343 in³
Answer: 343 in³
Explanation: In this problem, we're asked to find the volume of a cube.
It's important to understand that a cube is a type of rectangular prism and the formula for the volume of a rectangular prism is shown below.
Volume = length × width × height
In a cube however, the length, width, and height are all the same. So we can use the formula side × side × side instead.
So the formula for the volume of a cube is side × side × side or s³.
So to find the volume of the given cube, since each side has a length of 7 inches, we can plug this information into the formula to get (7 in.)³ or (7 in.)(7 in.)(7 in).
7 x 7 is 49 and 49 x 7 is 343.
So we have 343 in³.
So the volume of the given cube is 343 in³.
Answer:
343 in³
Step-by-step explanation:
Use the function f(x) = x2 + 6x + 6 and the graph of g(x) to determine the difference between the maximum value of g(x) and the minimum value of f(x). a parabola that opens down and passes through 0 comma 3, 3 comma 12, and 5 comma 8 15 12 9 3
The answer is 15,
because F(X)'s minimum value is (-3, -3)
and G(X)'s maximum value is (12, 3)
So when you subtract -3 from 12, you get a number that is greater then both. (because of the negative 3)
12 - (-3) = 15
Answer:
A. 15
Step-by-step explanation:
I got it rigth on the test :)
Have a great day fam!
Helppp i need answer asap
Answer:
Step-by-step explanation:
Triangle ABC is a right angle triangle.
From the given right angle triangle
BC represents the hypotenuse of the right angle triangle.
With m∠B as the reference angle,
AB represents the adjacent side of the right angle triangle.
AC represents the opposite side of the right angle triangle.
To determine BC , we would apply the Sine trigonometric ratio which is expressed as
Tan θ = opposite side/hypotenuse. Therefore,
Sin 44 = 10/BC
0.695 = 10/BC
BC = 10/0.695
BC = 14.4 to 1 decimal place.
A jar contains only pennies, nickels, dimes, and quarters. There are 14 pennies, 28 dimes, and 16 quarters. The rest of the coins are nickels. There are 70 coins in all. How many of the coins are not nickels? If n represents the number of nickels in the jar, what equation could you use to find n?
Answer:
Step-by-step explanation:
Let n represent the number of nickels in the jar.
The jar contains only pennies, nickels, dimes, and quarters. There are 14 pennies, 28 dimes, and 16 quarters. This means that the total number if pennies, dimes and quarters would be
14 + 28 + 16 = 58
This means that the number of coins that are not nickel is 58
The total number of coins in the jar is 70. Therefore, the equation that can be used to find n is
n + 58 = 70
n = 70 - 58
n = 12 nickels