Question 1

Solve the problem.
-When the temperature stays the same, the volume of a gas is inversely proportional to the pressure of the gas. If a balloon is filled with 204 cubic inches of a gas at a pressure of 14 pounds per square inch, find the new
Pressure of the gas if the volume is decreased to 68 cubic inches.
A) 39 pounds per square inch
B) 28 pounds per square inch
C) 42 pounds per square inch
D) $$\frac { 34 } { 7 }$$
Pounds per square inch

Accepted Answer

Since the volume of a gas is inversely proportional to the pressure when the temperature is constant (Boyle's Law), we can set up the equation $P_1V_1 = P_2V_2$, where $P_1$ and $V_1$ are the initial pressure and volume, and $P_2$ and $V_2$ are the final pressure and volume. Plugging in the given values, $14 \times 204 = P_2 \times 68$. Solving for $P_2$ gives $P_2 = \frac{14 \times 204}{68} = 42$ pounds per square inch.

Question 2

Solve.
-The amount of gas that a helicopter uses is directly proportional to the number of hours spent flying. The helicopter flies for 4 hours and uses 28 gallons of fuel. Find the number of gallons of fuel that the helicopter uses
To fly for 6 hours.
A) 24 gallons
B) 49 gallons
C) 48 gallons
D) 42 gallons

Accepted Answer

The helicopter uses 28 gallons for 4 hours, so it uses 28/4 = 7 gallons per hour. For 6 hours, it will use 7 * 6 = 42 gallons.

Question 3

Solve the problem.
-Two boats leave a dock at the same time. One boat is headed directly east at a constant speed of 35 knots
(nautical miles per hour), and the other is headed directly south at a constant speed of 22 knots. Express the
distance d between the boats as a function of the time t.

Accepted Answer

The answer of Solve the problem.
-Two boats leave a dock...

Question 4

Solve.
-If the resistance in an electrical circuit is held constant, the amount of current flowing through the circuit varies directly with the amount of voltage applied to the circuit. When 5 volts are applied to a circuit,
125 milliamperes of current flow through the circuit. Find the new current if the voltage is increased to 12 volts.
A) 325 milliamperes
B) 60 milliamperes
C) 288 milliamperes
D) 300 milliamperes

Accepted Answer

The relationship is direct variation, so I = kV. Given I = 125 mA when V = 5 volts, k = 125/5 = 25. For V = 12 volts, I = 25 * 12 = 300 mA.

Question 5

Solve the problem.
-The gravitational attraction A between two masses varies inversely as the square of the distance between them. The force of attraction is 2.25 lb when the masses are 4 ft apart, what is the attraction when the masses are 6 ft
Apart?
A) 1 lb
B) 2 lb
C) 4 lb
D) 3 lb

Accepted Answer

The gravitational attraction between two masses is given by the formula $A = \frac{k}{d^2}$, where $A$ is the attraction, $k$ is a constant, and $d$ is the distance between the masses. Given that the attraction is 2.25 lb when the distance is 4 ft, we can find $k$ by rearranging the formula to $k = A \cdot d^2$, thus $k = 2.25 \cdot 4^2 = 2.25 \cdot 16 = 36$. To find the attraction when the distance is 6 ft, we use the formula again with $d = 6$: $A = \frac{36}{6^2} = \frac{36}{36} = 1$ lb.

Question 6

If y varies directly as x, find a linear function which relates them.
-$$y = 3 \text { when } x = \frac { 1 } { 4 }$$ 
A) $f ( x ) = x + \frac { 11 } { 4 }$
B) $f ( x ) = \frac { x } { 3 }$
C) $f ( x ) = 12 x$
D) $f ( x ) = \frac { x } { 12 }$

Accepted Answer

Since $y$ varies directly as $x$, the relationship can be expressed as $y = kx$, where $k$ is the constant of variation. Given $y = 3$ when $x = \frac{1}{4}$, we can find $k$ by substituting these values into the equation: $3 = k \cdot \frac{1}{4}$, which gives $k = 12$. Therefore, the linear function that relates $y$ and $x$ is $y = 12x$, which corresponds to option C.

Question 7

Solve.
-The amount of water used to take a shower is directly proportional to the amount of time that the shower is in use. A shower lasting 15 minutes requires 7.5 gallons of water. Find the amount of water used in a shower
Lasting 8 minutes.
A) 120 gallons
B) 3.5 gallons
C) 4 gallons
D) 4.5 gallons

Accepted Answer

The amount of water used is directly proportional to the amount of time that the shower is in use. So, the ratio of water used to time is constant.For a shower lasting 15 minutes, the ratio is 7.5 gallons / 15 minutes = 0.5 gallons / minute.For a shower lasting 8 minutes, the amount of water used is 0.5 gallons / minute * 8 minutes = 4 gallons.

Question 8

If y varies directly as x, find a linear function which relates them.
-y = 24 when x = 9 
A) $f ( x ) = \frac { 8 } { 3 } x$
B) $f ( x ) = 3 x$
C) $f ( x ) = x + 15$
D) $f ( x ) = \frac { 3 } { 8 } x$

Accepted Answer

Since y varies directly as x, the relationship between them can be expressed as $y = kx$, where $k$ is the constant of variation. Given $y = 24$ when $x = 9$, we can find $k$ by substituting these values into the equation: $24 = k \cdot 9$, which gives $k = \frac{24}{9} = \frac{8}{3}$. Therefore, the linear function that relates them is $f(x) = \frac{8}{3}x$.

Question 9

If y varies directly as x, find a linear function which relates them.
-y = 0.2 when x = 1.6
A) f(x) = 8x
B) f(x) = 0.125x
C) f(x) = 0.2x
D) f(x) = x - 1.4

Accepted Answer

When y varies directly as x, the relationship can be described by the equation y = kx, where k is the constant of variation. Given y = 0.2 when x = 1.6, we can find k by substituting these values into the equation: 0.2 = k(1.6). Solving for k gives k = 0.2 / 1.6 = 0.125. Therefore, the linear function that relates them is f(x) = 0.125x.

Question 10

If y varies directly as x, find a linear function which relates them.
-y = 8 when x = 24 
A) $f ( x ) = \frac { x } { 3 }$
B) $f ( x ) = x + 16$
C) $f ( x ) = \frac { x } { 8 }$
D) $f ( x ) = 3 x$

Accepted Answer

When y varies directly as x, the relationship can be described by the equation $y = kx$, where $k$ is the constant of variation. Given $y = 8$ when $x = 24$, we can find $k$ by solving $8 = k(24)$, which gives $k = \frac{1}{3}$. Therefore, the linear function is $f(x) = \frac{x}{3}$.

Question 11

If y varies directly as x, find a linear function which relates them.
-y = 5.6 when x = 0.7
A) f(x) = x + 4.9
B) f(x) = 8x
C) f(x) = 0.7x
D) f(x) = 0.125x

Accepted Answer

Since y varies directly as x, the relationship between them can be represented by the equation y = kx, where k is the constant of variation. Given y = 5.6 when x = 0.7, we can find k by substituting these values into the equation: 5.6 = k(0.7). Solving for k gives k = 5.6 / 0.7 = 8. Therefore, the linear function that relates y and x is y = 8x, which corresponds to choice B.

Question 12

Solve the problem.
-If the force acting on an object stays the same, then the acceleration of the object is inversely proportional to its mass. If an object with a mass of 18 kilograms accelerates at a rate of 7 meters per second per second by a force,
Find the rate of acceleration of an object with a mass of 9 kilograms that is pulled by the same force.
Meters per second per second
A) $$\frac { 7 } { 2 }$$
B) 12 meters per second per second
C) 14 meters per second per second
D) 7 meters per second per second

Accepted Answer

Since acceleration is inversely proportional to mass when force is constant, halving the mass (from 18 kg to 9 kg) doubles the acceleration from 7 m/s^2 to 14 m/s^2.

Question 13

Solve the problem.
-If the voltage, V, in an electric circuit is held constant, the current, I, is inversely proportional to the resistance, R. If the current is 280 milliamperes when the resistance is 5 ohms, find the current when the resistance is 35 ohms.
A) 200 milliamperes
B) 40 milliamperes
C) 1953 milliamperes
D) 1960 milliamperes

Accepted Answer

The relationship between current (I), voltage (V), and resistance (R) is given by Ohm's law, V = IR. If the current is inversely proportional to the resistance while the voltage is constant, then I1R1 = I2R2. Given I1 = 280 milliamperes and R1 = 5 ohms, for R2 = 35 ohms, I2 = (I1R1)/R2 = (280*5)/35 = 40 milliamperes.

Question 14

Solve the problem.
-The amount of time it takes a swimmer to swim a race is inversely proportional to the average speed of the swimmer. A swimmer finishes a race in 150 seconds with an average speed of 4 feet per second. Find the
Average speed of the swimmer if it takes 200 seconds to finish the race.
A) 4 feet per second
B) 3 feet per second
C) 5 feet per second
D) 2 feet per second

Accepted Answer

The relationship between time and speed is given by $T = \frac{k}{S}$, where $T$ is the time, $S$ is the speed, and $k$ is a constant. Given $T_1 = 150$ seconds and $S_1 = 4$ feet per second, we find $k = T_1 \times S_1 = 150 \times 4 = 600$. For $T_2 = 200$ seconds, using $k = T_2 \times S_2$, we find $S_2 = \frac{k}{T_2} = \frac{600}{200} = 3$ feet per second.

Question 15

Solve the problem.
-A rocket is shot straight up in the air from the ground at a rate of 62 feet per second. The rocket is tracked by a range finder that is 450 feet from the launch pad. Let d represent the distance from the rocket to the range finder
And t represent the time, in seconds, since "blastoff". Express d as a function of t. 
A) $\mathrm { d } ( \mathrm { t } ) = 450 + 62 \mathrm { t } ^ { 2 }$
B) $d ( t ) = 450 ^ { 2 } + ( 62 t ) ^ { 2 }$
C) $d ( t ) = \sqrt { 62 ^ { 2 } + ( 450 t ) ^ { 2 } }$
D) $d ( t ) = \sqrt { 450 ^ { 2 } + ( 62 t ) ^ { 2 } }$

Accepted Answer

The distance from the rocket to the range finder can be found using the Pythagorean theorem, where the vertical distance the rocket has traveled is $62t$ and the horizontal distance to the range finder is 450 feet.

Question 16

Solve the problem.
-While traveling at a constant speed in a car, the centrifugal acceleration passengers feel while the car is turning is inversely proportional to the radius of the turn. If the passengers feel an acceleration of 16 feet per second per
Second when the radius of the turn is 60 feet, find the acceleration the passengers feel when the radius of the
Turn is 240 feet.
A) 4 feet per second per second
B) 7 feet per second per second
C) 5 feet per second per second
D) 6 feet per second per second

Accepted Answer

The acceleration is inversely proportional to the radius, so if the radius quadruples (from 60 to 240 feet), the acceleration is divided by 4, making it 16/4 = 4 feet per second per second.

Quiz 1: Functions and Their Graphs

Solve. -The amount of gas that a helicopter uses is directly proportional to the number of hours spent flying. The helicopter flies for 4 hours and uses 28 gallons of fuel. Find the number of gallons of fuel that the helicopter uses To fly for 6 hours.

Solve the problem. -Two boats leave a dock at the same time. One boat is headed directly east at a constant speed of 35 knots (nautical miles per hour), and the other is headed directly south at a constant speed of 22 knots. Express the distance d between the boats as a function of the time t.

Solve the problem. -The gravitational attraction A between two masses varies inversely as the square of the distance between them. The force of attraction is 2.25 lb when the masses are 4 ft apart, what is the attraction when the masses are 6 ft Apart?

If y varies directly as x, find a linear function which relates them. - $y = 3 \text { when } x = \frac { 1 } { 4 }$

Solve. -The amount of water used to take a shower is directly proportional to the amount of time that the shower is in use. A shower lasting 15 minutes requires 7.5 gallons of water. Find the amount of water used in a shower Lasting 8 minutes.

If y varies directly as x, find a linear function which relates them. -y = 24 when x = 9

If y varies directly as x, find a linear function which relates them. -y = 0.2 when x = 1.6

If y varies directly as x, find a linear function which relates them. -y = 8 when x = 24

If y varies directly as x, find a linear function which relates them. -y = 5.6 when x = 0.7

Solve the problem. -If the voltage, V, in an electric circuit is held constant, the current, I, is inversely proportional to the resistance, R. If the current is 280 milliamperes when the resistance is 5 ohms, find the current when the resistance is 35 ohms.