AP Calculus AB / Contextual Applications of Differentiation / Question No. 27

27. People are entering a museum at a rate modeled by \( f(t) \) people per hour and exiting the building at a rate modeled by \( g(t) \) people per hour, where \( t \) is measured in hours. The functions \( f \) and \( g \) are nonnegative and differentiable for all times \( t \).

Which of the following inequalities indicates that the rate of change of the number of people in the building is decreasing at time \( t \)?

\( g(t) < 0 \)

\( g'(t) < 0 \)

\( f(t) - g(t) < 0 \)

\( f'(t) - g'(t) < 0 \)

Answer is: option4

\( f'(t) - g'(t) < 0 \)

Solution:

The rate at which people enter the museum is given by \( f(t) \).

The rate at which people exit the museum is given by \( g(t) \).

The rate of change of the number of people in the museum is:

\[ N'(t) = f(t) - g(t) \]

The question asks when this rate is decreasing, meaning we need to consider the derivative of \( N'(t) \), which is:

\[ N''(t) = f'(t) - g'(t) \]

To determine when the rate of change is decreasing, we need:

\[ f'(t) - g'(t) < 0 \]

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