Question

a small pump takes 4 more hours than a large pump to empty a city swimming pool. the rate for the large pump to drain the pool can be represented by the rational expression \\(\frac{1}{x}\\), where \\(x\\) is the time that it takes for the large pump to drain the pool.

a. write a simplified rational expression that describes the combined rate if both pumps are used to drain the pool.
b. what is the combined rate to drain the pool if it takes the larger pump 8 hours to drain the pool?

a. the simplified rational expression that describes the combined rate if both pumps are used to drain the pool is
(type your answer in factored form. use integers or fractions for any numbers in the expression.)

Explanation:

Express individual rates

$$
LATEXBLOCK0
$$

Combine and simplify rates

$$
LATEXBLOCK1
$$

Evaluate combined rate at \(x = 8\)

$$
LATEXBLOCK2
$$

Answer:

Question a

The simplified rational expression that describes the combined rate if both pumps are used to drain the pool is \(\frac{2x + 4}{x(x + 4)}\).

Question b

What is the combined rate to drain the pool if it takes the larger pump 8 hours to drain the pool?
The combined rate is \(\frac{5}{24}\) of the pool per hour.