Question

1. when temperature is held constant, the volume v of a gas is inversely proportional to the pressure p of the gas on its container. a pressure of 32 pounds per square inch results in a volume of 20 cubic feet. what is the pressure if the volume becomes 10 cubic feet?

1. the time t (in days) that it takes to harvest a field varies inversely with the number n of farm workers. a farmer can harvest his crop in 20 days with 7 farm workers. how long will it take to harvest the crop if he hires 10 farm workers?

Explanation:

Problem 17

Step1: Define inverse proportionality

If \( V \) is inversely proportional to \( P \), then \( V=\frac{k}{P} \) (where \( k \) is the constant of proportionality).

Step2: Find the constant \( k \)

Given \( P = 32 \) and \( V = 20 \), substitute into \( V=\frac{k}{P} \): \( 20=\frac{k}{32} \). Multiply both sides by 32: \( k = 20\times32=640 \).

Step3: Find \( P \) when \( V = 10 \)

Now, use \( V=\frac{k}{P} \) with \( V = 10 \) and \( k = 640 \): \( 10=\frac{640}{P} \). Solve for \( P \): \( P=\frac{640}{10}=64 \).

Step1: Define inverse proportionality

If \( t \) is inversely proportional to \( n \), then \( t=\frac{k}{n} \) (where \( k \) is the constant of proportionality).

Step2: Find the constant \( k \)

Given \( t = 20 \) and \( n = 7 \), substitute into \( t=\frac{k}{n} \): \( 20=\frac{k}{7} \). Multiply both sides by 7: \( k = 20\times7 = 140 \).

Step3: Find \( t \) when \( n = 10 \)

Now, use \( t=\frac{k}{n} \) with \( n = 10 \) and \( k = 140 \): \( t=\frac{140}{10}=14 \).

Answer:

The pressure is 64 pounds per square inch.

Problem 18