Question

modeling with rational functions
using an inverse variation function to solve problems
for a fixed amount of a gas at a constant temperature, the volume of the gas is inversely proportional to its pressure. at a pressure of 30 pounds per square inch (psi), a gas has a volume of 600 in³.
which function can be used to model the volume of the gas y, in cubic inches, when the pressure is x psi?
what would the volume of the gas be if the pressure is increased to 40 psi?
if the volume of the gas increases from 600 to 800 in³, by how much does the pressure change?

Explanation:

First Sub - Question (Function Modeling)

Step1: Recall Inverse Variation Formula

Inverse variation is modeled by \( y=\frac{k}{x} \) (or \( y=\frac{k}{z} \) here, since pressure is \( z \)). We know when \( z = 30 \) psi, \( y=600\) in³. Substitute into the formula to find \( k \).
\( 600=\frac{k}{30} \)

Step2: Solve for k

Multiply both sides by 30: \( k = 600\times30=18000 \). So the function is \( y=\frac{18000}{z} \) (the option \( y = 18000x \) seems to have a typo, likely \( y=\frac{18000}{x} \) where \( x \) is pressure \( z \)).

Step1: Use the Inverse Variation Function

We have the function \( y=\frac{18000}{z} \). Now, \( z = 40 \) psi. Substitute \( z = 40 \) into the function.
\( y=\frac{18000}{40} \)

Step2: Calculate the Volume

\( \frac{18000}{40}=450 \) in³.

Step1: Find Initial Pressure

When \( y = 600 \) in³, \( z = 30 \) psi (given).

Step2: Find New Pressure when \( y = 800 \) in³

Use the function \( y=\frac{18000}{z} \). Substitute \( y = 800 \): \( 800=\frac{18000}{z} \). Solve for \( z \): \( z=\frac{18000}{800}=22.5 \) psi? Wait, no, wait. Wait, inverse variation: as volume increases, pressure decreases. Wait, initial \( y = 600 \), \( z = 30 \). New \( y = 800 \). So \( z=\frac{18000}{800}=22.5 \) psi. The change in pressure is \( 30 - 22.5=7.5 \) psi. So pressure decreases by 7.5 psi.

Step3: Determine the Change

Initial pressure \( P_1 = 30 \) psi, new pressure \( P_2=\frac{18000}{800}=22.5 \) psi. Change \( = P_1 - P_2=30 - 22.5 = 7.5 \) psi (decrease).

Answer:

\( y=\frac{18000}{x} \) (or the given option \( y = 18000x \) is likely a typo for \( y=\frac{18000}{x} \))

Second Sub - Question (Volume at 40 psi)