A water container is to be drained for cleansing. If Q represents the number of gallons of the water in the container t minutes after the container has started to drain, and Q(t) = 1/4 (50 − t)^2 .

Question

A water container is to be drained for cleansing. If Q represents the number of gallons of the water in the container t minutes after the container has started to drain, and Q(t) = 1/4 (50 − t)^2 .

a. Complete the following table

Q (Gallons): 625 576 529 484 - - - 324 289 256 225

t (min): 0 2 4 6 8 10 12 14 16 18 20

b. Estimate the instantaneous rate of change of Q at the instant t=8.

c. Write a sentence that clearly interprets the physical meaning of your answer in to part (b) in the context of the problem.

Answer 1

HI,

First I made a spreadsheet if Time and Q(t). Then I took the first derivation of Q in where I used the chain rule. So we get Q' = [1/4 * 2*(50-t)] * [-1] which reduces to Q' = -1/2(50-t). At t = 8, Q'(t=8) = -21.

The meaning of the answer at T = 8, is that the water is draining at that moment at 21 gallons/minute. (If it were positive, the container would be filling.). The rate at which the container is draining is steadily decreasing, that is it drains faster at first but then slows down as the amount of water decreases.

For this problem, you should review the chain rule: d f(g(x))/dx = f'(g(x))*g'(x). In our case, g(x) is (50-t), so f'(g(x)) = 1/4 * 2 *(50-t) and g'(x) = -1, giving us Q' = -1/2(50-t).