Question

The function h=f(t) 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 the next 10 minutes

a. What are the units of f' (t)​?
b. When are both f and f' the same value​?

Part 1
a. The units of f' (t)are
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Answer 1

Explanation:

a. The units of f'(t) are inches per minute.

In this context, f(t) represents the height of the water in the bathtub in inches at time t in minutes. So, when you take the derivative f'(t), you are calculating the rate of change of the height of the water with respect to time, which is how fast the height is changing. Since time is measured in minutes and the height is measured in inches, the units of f'(t) will be inches per minute.

Part 2

b. f(t) and f'(t) have the same value at points where the height of the water is changing at a constant rate.

In other words, when the water is being added to the bathtub at a constant rate (rising steadily), the value of f(t) and f'(t) will be the same. This means that at any time t during the period from t=2 (when you start filling the tub) to t=7 (when you turn off the water), f(t) and f'(t) will have the same value.

So, for t in the range [2, 7], f(t) and f'(t) have the same value because that's the time when the water is rising steadily in the tub.