Water leaks from a tank at the rate of r(t) gallons per hour. The rate decreased as time passed, and values of the rate at two-hour time intervals are shown in the table below

Question

Water leaks from a tank at the rate of r(t) gallons per hour. The rate decreased as time passed, and values of the rate at two-hour time intervals are shown in the table below. The total amount of water that leaked out is evaluated by a Riemann sum. Find the upper estimate (left end-points of each rectangle) for the total amount of water that leaked out by using five rectangles.

Give your answer with one decimal place

t(hr) r(t) (gal/hr)

0 10.7

2 8.6

4 6.6

6 5.2

8 5.0

10 4.5

Answer 1

The amount of water would be the integral of the rate function. The estimation using the Riemann sum and left endpoints:

F(x) ≅ ∑f(x_i-1)Δx where Δx=10/5 = 2 for this case and i = 1 to 5

W = (10.7 + 8.6 + 6.6 + 5.2 + 5.0)2 = 36.1•2 = 72.2 gallons