Calculus Bathtub Question

Question

Hello! I've been stuck on this calc problem and I could really use help:

The function H=f(f) gives the height of the water, in inches, in a bathtub at time t, in minutes. . Starting with an empty tub at t=0, you plug the drain, take off your watch, turn on some relaxing music, enter the tub, and then begin to fill it at t=2. The height of the water rises steadily until you turn off the water at t=7. You relax in the tub for 10 minutes until t=17, when you stand up to swat at a fly. you sit back down at at t=19 where the water rises as you sit back down.

A) What are the units of f'(t)?

a. Inches per minute, b. minutes, c. inches, d. minutes per inch

B) For which of the value(s) of t is f' negative?

a. t=4, b. t=8, c. t=19, d. t=17, e. t=1

Answer 1

Hello M T.! Let me make this understandable for you!

A) The units for the derivative f ' (x) is the unit of f (x) divided by the unit of x. In this question, the unit of h(t) is the height of the water (in INCHES), and the unit of t is in minutes. Therefore, according to the unit logic I just earlier explained, the units for h ' (t) is inches divided by minutes. The correct answer is (A) >>> (ANSWER)

B) In order to answer this question, we should understand what it means to have a negative derivative. According to the rules of calculus, when f(x) is decreasing, then f ' (x) must be negative. Therefore, if we want to know when f ' (t) < 0, then we need to know when the height of the ewater is decreasing.

Let's make a timeline of the water at various t's.

At 0 < t < 2, the water level was 0 and did not change, so f ' (t) = 0. Not negative, so not this time.

At 2 < t < 7, the water level rises steadily so f ' (t) > 0. Not negative.

At 7 < t < 17, the person in the bathtub relaxes and the water level did not increase. Not negative.

At t = 17, because that person stood up, the volume of the water displaced by the person's volume left the water so the height of the water in the bathtub decreased, so f ' (t) < 0. <<<(D) = ANSWER>>>

Well, at t = 19, the question even said that water level rised, fo f ' (t) > 0 and so it is not negative.

I hoped that helped you! Let me know if you have additional questions!