Question

this question has two parts. first, answer part a. then, answer part b.
part a
a swimming pool has 20,002\frac{2}{3} gallons of water and is being drained at a rate of 10\frac{1}{2} gallons per minute. another swimming pool is empty and is being filled at a rate of 20\frac{1}{4} gallons per minute. when will the swimming pools have the same amount of water?
write an equation to represent this situation, where x is the number of minutes.
part b
solve the equation. explain the meaning of the solution

Explanation:

Step1: Convert mixed numbers to fractions

First pool initial volume: $2002\frac{7}{8} = \frac{2002\times8 + 7}{8} = \frac{16023}{8}$ gallons
Drain rate: $10\frac{1}{2} = \frac{21}{2}$ gallons per minute
Fill rate: $20\frac{1}{4} = \frac{81}{4}$ gallons per minute

Step2: Set up volume equations

Volume of first pool after $x$ minutes: $\frac{16023}{8} - \frac{21}{2}x$
Volume of second pool after $x$ minutes: $0 + \frac{81}{4}x$
Set volumes equal:
$$\frac{16023}{8} - \frac{21}{2}x = \frac{81}{4}x$$

Step3: Combine like terms

Multiply all terms by 8 to eliminate denominators:
$16023 - 84x = 162x$
$16023 = 162x + 84x$
$16023 = 246x$

Step4: Solve for x

$$x = \frac{16023}{246} = 65.134... \approx 65$$
(Exact value: $\frac{16023}{246} = \frac{5341}{82} = 65\frac{11}{82}$)

Answer:

Part A Equation:

$\frac{16023}{8} - \frac{21}{2}x = \frac{81}{4}x$ (or simplified as $16023 - 84x = 162x$)

Part B Solution:

$x = \frac{5341}{82} \approx 65$ minutes. This means after approximately 65 minutes, the amount of water in both swimming pools will be equal.