Water flows into a storage tank at a rate given (in liters per minute) by the differentiable function r(t), where t is measured in minutes. Measurements of r at 5-minute intervals are recorded in the table below.
t (min)
0
5
10
15
20
25
30
r(t) (L/min)
9.5
8.7
7.9
7.1
6.4
5.9
5.3
At time t=0, the tank contains 500 liters of water.
(a) Use the trapezoidal rule with subintervals of length 5 minutes to approximate ∫030r(t)dt. Show all intermediate calculations.
(b) Use your result from part (a) to estimate the volume of water in the tank at t=30 minutes.
(c) Determine whether your estimate in part (b) is an overestimate or an underestimate of the actual volume. Justify your answer by analyzing the behavior of r(t) over the interval 0≤t≤30. Include a brief verbal explanation tied to the shape of the data.