Question 1

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-A student's grade on a test varies as the number of hours the student spends studying for the test.

Accepted Answer

The answer of Write the word or phrase that best...

Question 2

Solve the problem.
-The strength of a wooden beam varies jointly as the width of the beam and the square of the thickness of the beam, and inversely as the length of the beam. A beam that is 48 in. long, 6 in. wide, and 4 in. thick can support a load of $1,668 \mathrm { lb }$. Find the maximum load that can be safely supported by a board that is $8 \mathrm { in }$. wide, $120 \mathrm { in }$. long, and 5 in. thick.

Accepted Answer

The answer of Solve the problem.
-The strength of a wooden...

Question 3

Solve the problem.
-The yield on a bond varies inversely as the price. The yield on a particular bond is $3 \%$ when the price is $\$ 150$. What price is necessary for a yield of $4.5 \%$ ? Round to the nearest cent, if necessary.

Accepted Answer

The answer of Solve the problem.
-The yield on a bond...

Question 4

Solve the problem.
-$R$ varies directly as $x ^ { 2 }$ and inversely as $t$ when $R = 30$ when $x = 6$ and $t = 12$. Find $R$ when $x = 7$ and $t = 9$.

Accepted Answer

Given $R$ varies directly as $x^2$ and inversely as $t$, we use the formula $R = k\frac{x^2}{t}$ where $k$ is the constant of proportionality. Substituting the given values, $30 = k\frac{6^2}{12}$, we find $k = 10$. Then, for $x = 7$ and $t = 9$, $R = 10\frac{7^2}{9} = \frac{490}{9}$.

Question 5

Find the constant of variation k.
-$N$ varies jointly as $t$ and $p$. When $t$ is 6 and $p$ is $3 , N$ is 36 .

Accepted Answer

The formula for joint variation is $N = ktp$. Given $N = 36$, $t = 6$, and $p = 3$, substitute these values into the equation: $36 = k \times 6 \times 3$. Solving for $k$, we get $k = 36 / 18 = 2$.

Question 6

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The amount of a person's paycheck varies as the number of hours worked.

Accepted Answer

The answer of Write the word or phrase that best...

Question 7

Find the constant of variation k.
-p is inversely proportional to q. When q is 8, p is 3.

Accepted Answer

In an inverse variation, $ p \times q = k $. Given $ p = 3 $ and $ q = 8 $, we have $ 3 \times 8 = 24 $. Thus, $ k = 24 $.

Question 8

Choose the one alternative that best completes the statement or answers the question.
Write a variation model using k as the constant of variation.
-The variable s varies indirectly as t.

Accepted Answer

When a variable varies indirectly as another, it means the product of the two variables is constant. Therefore, $s = \frac { k } { t }$ correctly represents an inverse variation where $k$ is the constant of variation.

Question 9

Choose the one alternative that best completes the statement or answers the question.
Write a variation model using k as the constant of variation.
-The variable v varies jointly as w and x and inversely as the fourth root of y.

Accepted Answer

The correct model for joint variation with $w$ and $x$, and inverse variation with the fourth root of $y$, is $v = \frac { k w x } { \sqrt [ 4 ] { y } }$. This captures both the direct proportionality to $w$ and $x$ and the inverse proportionality to the fourth root of $y$.

Question 10

Choose the one alternative that best completes the statement or answers the question.
Write a variation model using k as the constant of variation.
-The variable m is directly proportional to n.

Accepted Answer

Direct proportionality between two variables means that one variable is equal to the other variable multiplied by a constant. In this case, $m$ is directly proportional to $n$, which is represented by the equation $m = kn$.

Question 11

Solve the problem.
-The amount of a pain reliever that a physician prescribes for a child varies directly as the weight of the child. A physician prescribes $170 \mathrm { mg }$ of the medicine for a 20-lb child.
a. How much medicine would be prescribed for a 40-b child?
b. If $140 \mathrm { mg }$ of medicine was prescribed, what is the weight of the child?

Accepted Answer

The amount of medicine prescribed varies directly with the weight of the child, so we use the formula $ \frac{170 \, \text{mg}}{20 \, \text{lb}} = \frac{x \, \text{mg}}{40 \, \text{lb}} $ for part a, solving for $x$ gives $340 \, \text{mg}$. For part b, using the direct variation formula $ \frac{140 \, \text{mg}}{x \, \text{lb}} = \frac{170 \, \text{mg}}{20 \, \text{lb}} $, solving for $x$ gives $16.47 \, \text{lb}$.

Question 12

Choose the one alternative that best completes the statement or answers the question.
Write a variation model using k as the constant of variation.
-The variable n is directly proportional to the fifth power of $ \alpha $ and inversely proportional to the fifth power of z.

Accepted Answer

Direct proportionality to $ \alpha^5 $ and inverse proportionality to $ z^5 $ is correctly represented by $ n = \frac { k \alpha ^ { 5 } } { z ^ { 5 } }$.

Question 13

Find the constant of variation k.
-y varies directly as x. When x is 16, y is 20.

Accepted Answer

The constant of variation $k$ is found using the formula $y = kx$. Substituting $x = 16$ and $y = 20$, we get $20 = 16k$, which simplifies to $k = \frac{20}{16} = \frac{5}{4}$.

Question 14

Solve the problem.
-The period of a pendulum is the length of time required to complete one swing back and forth. The period varies directly as the square root of the length of the pendulum. If it takes $1.6 \mathrm { sec }$ for a $0.64 \mathrm {~m}$ pendulum to complete one period, what is the period of a $1.21 \mathrm {~m}$ pendulum?

Accepted Answer

The answer of Solve the problem.
-The period of a pendulum...

Question 15

Solve the problem.
-The area of a picture projected on a wall varies directly as the square of the distance from the projector to the wall.
a. If a $12 - \mathrm { ft }$ distance produces a $81 \mathrm { ft } ^ { 2 }$ picture, what is the area of the picture when the projection unit is moved to a distance of $16 \mathrm { ft }$ from the wall?
b. If the projected image is $324 \mathrm { ft } ^ { 2 }$, how far is the projector from the wall?

Accepted Answer

The area of the picture varies directly as the square of the distance from the projector to the wall, so we use the formula $A = kD^2$, where $A$ is the area, $D$ is the distance, and $k$ is the constant of variation. For part a, we find $k$ using the given values and then calculate the area for a 16 ft distance. For part b, we solve for $D$ given $A = 324 \, \text{ft}^2$.

Question 16

Solve the problem.
-The resistance of a wire varies directly as its length and inversely as the square of its diameter. A 50 -ft wire with a $0.4$-in. diameter has a resistance of $0.0125 \Omega$. Find the resistance of a 20 -ft wire with a diameter of $0.1$-in. Round to 4 decimal places if necessary.

Accepted Answer

The resistance of a wire varies directly as its length and inversely as the square of its diameter. So, the resistance of a wire can be expressed as R = k(L/d^2), where k is a constant. We can find the value of k by using the given information: R = 0.0125 ohms, L = 50 ft, and d = 0.4 in. Substituting these values into the equation, we get: 0.0125 = k(50/0.4^2). Solving for k, we get: k = 0.000025. Now we can use this value of k to find the resistance of a 20-ft wire with a diameter of 0.1 in: R = k(L/d^2) = 0.000025(20/0.1^2) = 0.08 ohms.

Question 17

Solve the problem.
-The number of hours it takes to paint a house is inversely proportional to the number of people painting. If it takes 3 workers $17.7$ hours to paint a certain house, how long would it take 5 workers? Round to one tenth of an hour.

Accepted Answer

The time it takes to paint a house is inversely proportional to the number of workers, so $T = k/N$, where $T$ is time, $N$ is the number of workers, and $k$ is a constant. With 3 workers taking 17.7 hours, $k = 3 \times 17.7 = 53.1$. For 5 workers, $T = 53.1 / 5 = 10.62$, which rounds to $10.6$ hours.

Question 18

Solve the problem.
-The body mass index (BMI) of an individual varies directly as the weight of the individual and inversely as the square of the height of the individual. The body mass index for a 154 -lb person who is $71 \mathrm { in }$. tall is $21.48$. Determine the BMI for an individual who is $77 \mathrm { in }$. tall and $164 \mathrm { lb }$. Round to 2 decimal places.

Accepted Answer

The answer of Solve the problem.
-The body mass index (BMI)...

Question 19

Solve the problem.
-The cost, in dollars, of filling your gas tank is directly proportional to the amount purchased. If 11 gallons of gas costs $\$ 13.20$, how much would 14 gallons cost?

Accepted Answer

The cost is directly proportional to the amount purchased, so the cost per gallon is $13.20 / 11 = $1.20. For 14 gallons, the cost would be 14 * $1.20 = $16.80.

Question 20

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The volume of a right circular cone varies as the square of the radius of the cylinder
and as the height of the cylinder.

Accepted Answer

The answer of Write the word or phrase that best...

Quiz 3: Polynomial and Rational Functions

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A student's grade on a test varies as the number of hours the student spends studying for the test.

Solve the problem. -The yield on a bond varies inversely as the price. The yield on a particular bond is $3 \%$ when the price is $\$ 150$ . What price is necessary for a yield of $4.5 \%$ ? Round to the nearest cent, if necessary.

Solve the problem. - $R$ varies directly as $x ^ { 2 }$ and inversely as $t$ when $R = 30$ when $x = 6$ and $t = 12$ . Find $R$ when $x = 7$ and $t = 9$ .

Find the constant of variation k. - $N$ varies jointly as $t$ and $p$ . When $t$ is 6 and $p$ is $3 , N$ is 36 .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The amount of a person's paycheck varies as the number of hours worked.

Find the constant of variation k. -p is inversely proportional to q. When q is 8, p is 3.

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable s varies indirectly as t.

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable v varies jointly as w and x and inversely as the fourth root of y.

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable m is directly proportional to n.

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable n is directly proportional to the fifth power of $\alpha$ and inversely proportional to the fifth power of z.

Find the constant of variation k. -y varies directly as x. When x is 16, y is 20.

Solve the problem. -The number of hours it takes to paint a house is inversely proportional to the number of people painting. If it takes 3 workers $17.7$ hours to paint a certain house, how long would it take 5 workers? Round to one tenth of an hour.

Solve the problem. -The cost, in dollars, of filling your gas tank is directly proportional to the amount purchased. If 11 gallons of gas costs $\$ 13.20$ , how much would 14 gallons cost?

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The volume of a right circular cone varies as the square of the radius of the cylinder and as the height of the cylinder.

Quiz 3: Polynomial and Rational Functions

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A student's grade on a test varies as the number of hours the student spends studying for the test.

Solve the problem. -The yield on a bond varies inversely as the price. The yield on a particular bond is 3%3 \%3% when the price is $150\$ 150$150 . What price is necessary for a yield of 4.5%4.5 \%4.5% ? Round to the nearest cent, if necessary.

Solve the problem. - RRR varies directly as x2x ^ { 2 }x2 and inversely as ttt when R=30R = 30R=30 when x=6x = 6x=6 and t=12t = 12t=12 . Find RRR when x=7x = 7x=7 and t=9t = 9t=9 .

Find the constant of variation k. - NNN varies jointly as ttt and ppp . When ttt is 6 and ppp is 3,N3 , N3,N is 36 .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The amount of a person's paycheck varies as the number of hours worked.

Find the constant of variation k. -p is inversely proportional to q. When q is 8, p is 3.

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable s varies indirectly as t.

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable v varies jointly as w and x and inversely as the fourth root of y.

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable m is directly proportional to n.

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable n is directly proportional to the fifth power of α \alpha α and inversely proportional to the fifth power of z.

Find the constant of variation k. -y varies directly as x. When x is 16, y is 20.

Solve the problem. -The number of hours it takes to paint a house is inversely proportional to the number of people painting. If it takes 3 workers 17.717.717.7 hours to paint a certain house, how long would it take 5 workers? Round to one tenth of an hour.

Solve the problem. -The cost, in dollars, of filling your gas tank is directly proportional to the amount purchased. If 11 gallons of gas costs $13.20\$ 13.20$13.20 , how much would 14 gallons cost?

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The volume of a right circular cone varies as the square of the radius of the cylinder and as the height of the cylinder.

Solve the problem. -The yield on a bond varies inversely as the price. The yield on a particular bond is $3 \%$ when the price is $\$ 150$ . What price is necessary for a yield of $4.5 \%$ ? Round to the nearest cent, if necessary.

Solve the problem. - $R$ varies directly as $x ^ { 2 }$ and inversely as $t$ when $R = 30$ when $x = 6$ and $t = 12$ . Find $R$ when $x = 7$ and $t = 9$ .

Find the constant of variation k. - $N$ varies jointly as $t$ and $p$ . When $t$ is 6 and $p$ is $3 , N$ is 36 .

Choose the one alternative that best completes the statement or answers the question. Write a variation model using k as the constant of variation. -The variable n is directly proportional to the fifth power of $\alpha$ and inversely proportional to the fifth power of z.

Solve the problem. -The number of hours it takes to paint a house is inversely proportional to the number of people painting. If it takes 3 workers $17.7$ hours to paint a certain house, how long would it take 5 workers? Round to one tenth of an hour.

Solve the problem. -The cost, in dollars, of filling your gas tank is directly proportional to the amount purchased. If 11 gallons of gas costs $\$ 13.20$ , how much would 14 gallons cost?