Question

solve each inverse variation problem below

1. the volume ( v ) of a gas varies inversely as the pressure ( p ) on it. if the volume is ( 240 , \text{cm}^3 ) when the pressure is ( 30 , \text{kg/cm}^2 ), what pressure has to be applied to have a volume of ( 100 , \text{cm}^3 )?

Explanation:

Step1: Define inverse variation

For inverse variation, \( V \propto \frac{1}{P} \), so \( V = \frac{k}{P} \) (where \( k \) is the constant of variation).

Step2: Find the constant \( k \)

Given \( V = 240 \, \text{cm}^3 \) and \( P = 30 \, \text{kg/cm}^2 \), substitute into \( V = \frac{k}{P} \):
\( 240 = \frac{k}{30} \)
Multiply both sides by 30: \( k = 240 \times 30 = 7200 \).

Step3: Find new pressure \( P \)

We need \( V = 160 \, \text{cm}^3 \) and \( k = 7200 \). Use \( V = \frac{k}{P} \):
\( 160 = \frac{7200}{P} \)
Multiply both sides by \( P \): \( 160P = 7200 \)
Divide by 160: \( P = \frac{7200}{160} = 45 \, \text{kg/cm}^2 \).

Answer:

The required pressure is \( \boldsymbol{45 \, \text{kg/cm}^2} \).