Question: The National Air and Space Museum, in Washington D.C., is the most visited museum in the world. The rate at which visitors arrive at the Air and Space Museum on a busy summer day can be modeled by the function \(A\) defined by:

The rate at which people leave the museum on the same day can be modeled by the function \(L\) defined by

Both \(A(t)\) and \(L(t)\) are measured in people per hour and time \(t\) is measured in hours after midnight. These functions are valid for \(10 \leq t \leq 17.5\), the hours during which the museum is open. At time \(t=10\), there are no visitors in the museum.

1. How many people have arrived in the museum by noon? Round your answer to the nearest whole number.
2. Let \(F(t)=\int_{10}^{t} A(x)-L(x) d x\) for \(10 \leq t \leq 17.5\). Find the values of \(F(15)\) and
   \(F^{\prime}(15)\), and explain the meaning of \(F(15)\) and \(F^{\prime}(15)\) in the context of the museum.
3. At what time \(\mathrm{t}\), for \(10 \leq t \leq 17.5\), does the model predict that the number of visitors in the museum is a maximum?
4. Admission to the museum is free, but roughly 50% of the visitors who enter the museum between noon and 1 o'clock see the Imax movie, at $8.50 person, and roughly 20% of the visitors who enter the museum between noon and 1 o'clock see the planetarium show, at $6.50 person. Approximately how much money does the museum collect on this day from the visitors who arrive between noon and 1 o'clock and see either the movie or the planetarium show?

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