Use this piecewise function that gives the rate

Question

Use this piecewise function that gives the rate, r(t), in gallons per hour, at which a pool is being filled with water at time t hours. The pool is initially empty.

Based on this rate function, how many gallons of water are in the pool after 10 hours?

2,500

3,500

4,500

5,500

Answer 1

We have to use the integral of the function in those intervals:

V = integral from 0 to 2 of 250t + integral from 2 to 10 of 500

= [250 (t²/2) from 0 to 2 ] + [500t from 2 to 10]

= 250 (4/2 - 0) + 500(10 - 2)

= 500 + 4000

= 4500 Gallons