Question 1

Determine the constant of variation for the stated condition.
-$y$ varies jointly as $x$ and $z$. $y = 7.5$ when $x = 75$ and $z = 10$. Find $y$ when $x = 55$ and $z = 10$.
A) $5.5$
B) 550
C) 55
D) 11

Accepted Answer

The constant of variation $k$ can be found using the formula $y = kxz$ by substituting the given values: $7.5 = k \cdot 75 \cdot 10$. Solving for $k$, we find $k = 0.01$. To find $y$ when $x = 55$ and $z = 10$, we use the formula with the found $k$: $y = 0.01 \cdot 55 \cdot 10 = 5.5$.

Question 2

Determine the constant of variation for the stated condition.
-f varies jointly as $q ^ { 2 }$ and $h$, and $f = 64$ when $q = 4$ and $h = 2$. Find $q$ when $f = 90$ and $h = 5$.
A) $q = 3$
B) $q = 2$
C) $\mathrm { q } = 4$
D) $q = 5$ q

Accepted Answer

The constant of variation $k$ can be found using the equation $f = kq^2h$. Substituting the given values, $64 = k(4)^2(2)$, we find $k = 2$. To find $q$ when $f = 90$ and $h = 5$, we use the equation $90 = 2q^2(5)$. Solving for $q$, we find $q = 3$.

Question 3

Determine the constant of variation for the stated condition.
-While traveling in a car, the centrifugal force a passenger experiences as the car drives in a circle varies jointly as the mass of the passenger and the square of the speed of the car. If a passenger experiences a force of 57.6 newtons when the car is moving at a speed of 40 kilometers per hour and the passenger has a mass of 40 kilograms, find the force a passenger experiences when the car is moving at 30 kilometers per hour and the passenger has a mass of 70 kilograms.
A) 56.7 newtons
B) 63 newtons
C) 50.4 newtons
D) 72 newtons

Accepted Answer

The constant of variation is 0.09. The force experienced by the passenger is 56.7 newtons.

Question 4

Determine the constant of variation for the stated condition.
-The power that a resistor must dissipate is jointly proportional to the square of the current flowing through the resistor and the resistance of the resistor. If a resistor needs to dissipate 28 watts of power when 2 amperes of current is flowing through the resistor whose resistance is 7 ohms, find the power that a resistor needs to dissipate when 3 amperes of current are flowing through a resistor whose resistance is 7 ohms.
A) 63 watts
B) 21 watts
C) 147 watts
D) 42 watts

Accepted Answer

The power dissipated by a resistor, according to the given condition, is directly proportional to the square of the current (I^2) and the resistance (R). The formula for power (P) is P = k*I^2*R, where k is the constant of proportionality. Given P = 28 watts, I = 2 amperes, and R = 7 ohms, we can find k by rearranging the formula: 28 = k*(2^2)*7. Solving for k gives us k = 1. To find the power dissipated with 3 amperes of current and the same resistance, we use P = 1*(3^2)*7 = 63 watts.

Question 5

Determine the constant of variation for the stated condition.
-The amount of simple interest earned on an investment over a fixed amount of time is jointly proportional to the principle invested and the interest rate. A principle investment of $2000.00 with an interest rate of 1% earned $100.00 in simple interest. Find the amount of simple interest earned if the principle is $3500.00 and the interest rate is 8%.
A) $1400.00
B) $140,000.00
C) $175.00
D) $800.00

Accepted Answer

The constant of variation (k) can be found using the formula for simple interest: I = PRT, where I is the interest earned, P is the principal, R is the rate, and T is the time. Given that T is constant and not provided, we can assume it to be part of the constant of variation. From the given information, $100 = $2000 * 1% * k, solving for k gives us k = 1 (since 1% is 0.01, and 100 = 2000 * 0.01 * k). For a principal of $3500 and a rate of 8%, the interest earned would be I = $3500 * 8% * 1 = $280. However, the correct calculation should be based on the formula without assuming k because the time factor is implicitly considered constant. The correct calculation is I = PRT without needing to solve for k, which was incorrectly approached in the explanation. The correct interest for $3500 at 8% is actually $280, indicating a mistake in the initial calculation and choice selection. The correct approach to find the interest without the unnecessary step of finding k would directly apply the given formula: I = PRT, leading to the realization that none of the provided options match the correct calculation of $280, suggesting a reevaluation of the answer choices or an error in the question's setup or answer options.

Question 6

Determine the constant of variation for the stated condition.
-The amount of paint needed to cover the walls of a room varies jointly as the perimeter of the room and the height of the wall. If a room with a perimeter of 70 feet and 10 -foot walls requires 7 quarts of paint, find the amount of paint needed to cover the walls of a room with a perimeter of 55 feet and 8 -foot walls.
A) $4.4$ quarts
B) 440 quarts
C) 44 quarts
D) $8.8$ quarts

Accepted Answer

The constant of variation (k) can be found using the formula for joint variation, $y = kxz$, where y is the amount of paint, x is the perimeter, and z is the height of the walls. Given y = 7 quarts, x = 70 feet, and z = 10 feet, we solve for k: $7 = k(70)(10)$, resulting in $k = \frac{7}{700} = \frac{1}{100}$. To find the amount of paint needed for a perimeter of 55 feet and 8-foot walls, we use $y = \frac{1}{100}(55)(8) = 4.4$ quarts.

Question 7

Determine the constant of variation for the stated condition.
-$\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$, and $\mathrm { f } = 96$ when $\mathrm { q } = 4$ and $\mathrm { h } = 3$. Find $\mathrm { f }$ when $\mathrm { q } = 3$ and $\mathrm { h } = 4$.
A) $\mathrm { f } = 72$
B) $f = 24$
C) $\mathrm { f } = 18$
D) $\mathrm { f } = 8$

Accepted Answer

The constant of variation $k$ can be found using the given values: $f = kq^2h$. Substituting $f = 96$, $q = 4$, and $h = 3$ gives $96 = k(4)^2(3)$, solving for $k$ gives $k = 2$. To find $f$ when $q = 3$ and $h = 4$, substitute these values and $k = 2$ into the original equation: $f = 2(3)^2(4) = 72$.

Question 8

Determine the constant of variation for the stated condition.
-f varies jointly as $\mathrm { q } ^ { 2 }$ and $h$, and $\mathrm { f } = - 54$ when $\mathrm { q } = 3$ and $\mathrm { h } = 3$. Find $\mathrm { f }$ when $\mathrm { q } = 4$ and $\mathrm { h } = 5$.
A) $f = - 160$
B) $f = - 40$
C) $f = - 32$
D) $\mathrm { f } = - 10$

Accepted Answer

The formula for joint variation is $f = kq^2h$, where $k$ is the constant of variation. Substituting the given values, $f = -54$, $q = 3$, and $h = 3$, we get $-54 = k(3)^2(3)$. Solving for $k$, we find $k = -2$. To find $f$ when $q = 4$ and $h = 5$, substitute these values and $k = -2$ into the formula: $f = -2(4)^2(5) = -160$.

Question 9

Determine the constant of variation for the stated condition.
-The voltage across a resistor is jointly proportional to the resistance of the resistor and the current flowing through the resistor. If the voltage across a resistor is 32 volts for a resistor whose resistance is 4 ohms and when the current flowing through the resistor is 8 amperes, find the voltage across a resistor whose resistance is 2 ohms and when the current flowing through the resistor is 7 amperes.
A) 14 volts
B) 28 volts
C) 56 volts
D) 16 volts

Accepted Answer

The constant of variation (k) can be found using the formula V = IR, where V is voltage, I is current, and R is resistance. For the first scenario, V = 32 volts, I = 8 amperes, and R = 4 ohms. Thus, k = V/(IR) = 32/(8*4) = 1. For the second scenario, with R = 2 ohms and I = 7 amperes, V = k * I * R = 1 * 7 * 2 = 14 volts.

Question 10

Determine the constant of variation for the stated condition.
-$\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$, and $\mathrm { f } = 96$ when $\mathrm { q } = 4$ and $\mathrm { h } = 2$. Find $h$ when $\mathrm { f } = 135$ and $\mathrm { q } = 3$.
A) $h = 5$
B) $h = 2$
C) $h = 4$
D) $h = 3$

Accepted Answer

The constant of variation $k$ can be found using the equation $f = kq^2h$. Substituting $f = 96$, $q = 4$, and $h = 2$ gives $96 = k(4)^2(2)$, leading to $k = 6$. Then, using $f = 135$, $q = 3$, and solving for $h$ with $k = 6$, we get $135 = 6(3)^2h$, which simplifies to $h = 5$.

Question 11

Determine the constant of variation for the stated condition.
-$c$ varies jointly as $a$ and $b$. Find $c$ when $a = 15 , b = 13$, and $k = 3$.
A) $\mathrm { c } = 585$
B) $\mathrm { c } = 195$
C) $\mathrm { c } = 65$
D) $\mathrm { c } = \frac { 13 } { 5 }$

Accepted Answer

The constant of variation $k = 3$, and $c$ varies jointly as $a$ and $b$, so $c = kab$. Substituting $a = 15$, $b = 13$, and $k = 3$ gives $c = 3 \times 15 \times 13 = 585$.

Question 12

Determine the constant of variation for the stated condition.
-The pressure of a gas varies jointly as the amount of the gas (measured in moles) and the temperature and inversely as the volume of the gas. If the pressure is 1170 kPa (kiloPascals) when the number of moles is 8, the temperature is 260° Kelvin, and the volume is 960 cc, find the pressure when the number of moles is 10, the temperature is 290° K, and the volume is 600 cc.
A) 2610
B) 2880
C) 1305
D) 1170

Accepted Answer

The constant of variation k can be found using the formula P = k(nT/V). First, solve for k using the given conditions: 1170 = k(8*260/960). Then use k to find the new pressure with the new conditions: P = k(10*290/600).

Quiz 3: Polynomial and Rational Functions

Determine the constant of variation for the stated condition. - $y$ varies jointly as $x$ and $z$ . $y = 7.5$ when $x = 75$ and $z = 10$ . Find $y$ when $x = 55$ and $z = 10$ .

Determine the constant of variation for the stated condition. -f varies jointly as $q ^ { 2 }$ and $h$ , and $f = 64$ when $q = 4$ and $h = 2$ . Find $q$ when $f = 90$ and $h = 5$ .

Determine the constant of variation for the stated condition. - $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = 96$ when $\mathrm { q } = 4$ and $\mathrm { h } = 3$ . Find $\mathrm { f }$ when $\mathrm { q } = 3$ and $\mathrm { h } = 4$ .

Determine the constant of variation for the stated condition. -f varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = - 54$ when $\mathrm { q } = 3$ and $\mathrm { h } = 3$ . Find $\mathrm { f }$ when $\mathrm { q } = 4$ and $\mathrm { h } = 5$ .

Determine the constant of variation for the stated condition. - $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = 96$ when $\mathrm { q } = 4$ and $\mathrm { h } = 2$ . Find $h$ when $\mathrm { f } = 135$ and $\mathrm { q } = 3$ .

Determine the constant of variation for the stated condition. - $c$ varies jointly as $a$ and $b$ . Find $c$ when $a = 15 , b = 13$ , and $k = 3$ .

Quiz 3: Polynomial and Rational Functions

Determine the constant of variation for the stated condition. - yyy varies jointly as xxx and zzz . y=7.5y = 7.5y=7.5 when x=75x = 75x=75 and z=10z = 10z=10 . Find yyy when x=55x = 55x=55 and z=10z = 10z=10 .

Determine the constant of variation for the stated condition. -f varies jointly as q2q ^ { 2 }q2 and hhh , and f=64f = 64f=64 when q=4q = 4q=4 and h=2h = 2h=2 . Find qqq when f=90f = 90f=90 and h=5h = 5h=5 .

Determine the constant of variation for the stated condition. -f varies jointly as q2\mathrm { q } ^ { 2 }q2 and hhh , and f=−54\mathrm { f } = - 54f=−54 when q=3\mathrm { q } = 3q=3 and h=3\mathrm { h } = 3h=3 . Find f\mathrm { f }f when q=4\mathrm { q } = 4q=4 and h=5\mathrm { h } = 5h=5 .

Determine the constant of variation for the stated condition. - f\mathrm { f }f varies jointly as q2\mathrm { q } ^ { 2 }q2 and hhh , and f=96\mathrm { f } = 96f=96 when q=4\mathrm { q } = 4q=4 and h=2\mathrm { h } = 2h=2 . Find hhh when f=135\mathrm { f } = 135f=135 and q=3\mathrm { q } = 3q=3 .

Determine the constant of variation for the stated condition. - ccc varies jointly as aaa and bbb . Find ccc when a=15,b=13a = 15 , b = 13a=15,b=13 , and k=3k = 3k=3 .

Determine the constant of variation for the stated condition. - $y$ varies jointly as $x$ and $z$ . $y = 7.5$ when $x = 75$ and $z = 10$ . Find $y$ when $x = 55$ and $z = 10$ .

Determine the constant of variation for the stated condition. -f varies jointly as $q ^ { 2 }$ and $h$ , and $f = 64$ when $q = 4$ and $h = 2$ . Find $q$ when $f = 90$ and $h = 5$ .

Determine the constant of variation for the stated condition. -f varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = - 54$ when $\mathrm { q } = 3$ and $\mathrm { h } = 3$ . Find $\mathrm { f }$ when $\mathrm { q } = 4$ and $\mathrm { h } = 5$ .

Determine the constant of variation for the stated condition. - $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = 96$ when $\mathrm { q } = 4$ and $\mathrm { h } = 2$ . Find $h$ when $\mathrm { f } = 135$ and $\mathrm { q } = 3$ .

Determine the constant of variation for the stated condition. - $c$ varies jointly as $a$ and $b$ . Find $c$ when $a = 15 , b = 13$ , and $k = 3$ .