Problem 48 - Introductory Algebra for College Students

Question

One pipe can fill a swimming pool in 2 hours, a second can fill the pool in 3 hours, and a third pipe can fill the pool in 4 hours. How many minutes, to the nearest minute, would it take to fill the pool with all three pipes operating?

Step-by-Step Solution

Answer

It would take approximately 55 minutes to fill the pool with all three pipes operating.

Step 1: Calculate the rate of each pipe

The first pipe can fill the pool in 2 hours, so it fills \( \frac{1}{2} \) of the pool in one hour. Similarly, the second pipe fills \( \frac{1}{3} \) of the pool in one hour, and the third pipe fills \( \frac{1}{4} \) of the pool in one hour.

Step 2: Calculate the combined rate

Add the rates to get the combined rate of all the pipes. This will be \( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12} \) pools per hour.

Step 3: Calculate the time it would take

To find the time it would take for all pipes to fill the pool, divide the total size (1 pool) by the combined rate. That's \( \frac{1}{\frac{13}{12}} = \frac{12}{13} \) hours. Since there are 60 minutes in 1 hour, multiply \( \frac{12}{13} \) by 60 to get approximately 55.38 minutes, which rounds to 55 minutes to the nearest minute.