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

a) 441, 400, 361

b) 1/2(50-t)=Q’(t)

Q’(8)=1/2(50-8)=21

c) 8 seconds after initial draining, the container will be losing 21 gallons per minute of water instantaneously.

Answer 2

For part (a) of this problem, this table is simply expressing the values of Q at times t. This means that you just need to substitute each value of t into the given equation to find the correct value. This should be straightforward with the use of pen and paper or a calculator.

Part (b) requires you to estimate the instantaneous rate of change at a certain time. As we know, instantaneous rate of change refers to the derivative. So, in order to find this value, you would need to take the derivative of the given equation and then, once you have that, you can substitute t = 8 in order to find the value of the derivative at that precise time.

Part (c) is checking your understanding of the derivative and how it relates to rate of change. To help you with this, you should think about the units that the derivative is in. This will help you formulate a logical sentence that clearly demonstrates the interpretation of the derivative at that time.