Question

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 in.³.
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 in.³, by how much does the pressure change?

Explanation:

Step1: Model the inverse proportionality

Since volume \( y \) is inversely proportional to pressure \( x \), the relationship is \( y=\frac{k}{x} \), where \( k \) is the constant of proportionality. We know when \( x = 30 \), \( y=600 \). Substitute these values into the equation: \( 600=\frac{k}{30} \). Solving for \( k \), we multiply both sides by 30: \( k = 600\times30=18000 \). So the function is \( y=\frac{18000}{x} \).

Step2: Find volume at 40 psi

Use the function \( y=\frac{18000}{x} \). Substitute \( x = 40 \): \( y=\frac{18000}{40}=450 \) cubic inches.

Step3: Find pressure change when volume increases to 800

First, find the pressure when \( y = 800 \). Use \( y=\frac{18000}{x} \), so \( 800=\frac{18000}{x} \). Solve for \( x \): \( x=\frac{18000}{800}=22.5 \) psi. The original pressure was 30 psi, so the change is \( 30 - 22.5 = 7.5 \) psi (decrease of 7.5 psi).

Answer:

1. The function is \( y=\frac{18000}{x} \).
2. The volume at 40 psi is \( 450 \) \( \text{in}^3 \).
3. The pressure decreases by \( 7.5 \) psi.