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The rate at which water leaks from a tank, in gallons per hour, is modeled by \( R \), a differentiable function of the number of hours after the leak is discovered. Which of the following is the best interpretation of \( R'(3) \)?

A. The amount of water, in gallons, that has leaked out of the tank during the first three hours after the leak is discovered.

B. The amount of change, in gallons per hour, in the rate at which water is leaking during the three hours after the leak is discovered.

C. The rate at which water leaks from the tank, in gallons per hour, three hours after the leak is discovered.

D. The rate of change of the rate at which water leaks from the tank, in gallons per hour per hour.

Answer :

The problem provides us with a differentiable function R that models the rate at which water leaks from a tank in gallons per hour, as a function of the number of hours after the leak is discovered. We are then asked to interpret R'(3), which means the derivative of R with respect to time evaluated at t=3.
The CORRECT option is C


Option A suggests that R'(3) represents the amount of water that has leaked out of the tank during the first three hours after the leak is discovered. This interpretation is incorrect, as R'(3) represents the rate of change of the water leakage, not the actual amount of water leaked.

Option B proposes that R'(3) represents the amount of change in gallons per hour of the rate at which water is leaking during the three hours after the leak is discovered. This interpretation is also incorrect, as the derivative R'(t) represents the instantaneous rate of change of the function R at time t, not the change over a specific interval.

Option C suggests that R'(3) represents the rate at which water leaks from the tank, in gallons per hour, three hours after the leak is discovered. This interpretation is correct. The derivative R'(t) gives us the rate of change of the function R at time t, and evaluating this at t=3 gives us the rate of water leakage at that specific time.

Option D proposes that R'(3) represents the rate of change of the rate at which water leaks from the tank, in gallons per hour per hour. This interpretation is incorrect, as the derivative of the rate of change of R would give us the second derivative of the function, not the first derivative evaluated at a specific time.

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