Free response with a graphing calculator (Part A)
Museum visitors: table and model
- Units 1, 4, 6 and 8
- 9 points
- About 15 minutes
You can use a calculator on this question, just like on exam day.
A multi-part problem, usually set in a real-world context, where you need a graphing calculator for things like definite integrals, derivatives at a point and solving equations. You show your setup, give decimal answers to three places, and explain or justify your conclusions. On the exam: 2 questions in Part A of the free-response section (30 minutes, graphing calculator required). The whole free-response section is 6 questions in 90 minutes and counts for 50% of the score. AB and BC use the same format; on BC, one of these is usually a parametric, polar or vector question.
The question and its sources
A museum is open from 9 a.m. (t = 0) to 5 p.m. (t = 8), where t is measured in hours. Visitors enter the museum at a rate modeled by a continuous function E(t), measured in people per hour. Selected values of E(t) are given in the table. During the same time interval, visitors leave the museum at a rate modeled by L(t) = 200(1 − e^(−t²/12)) people per hour. There are no visitors in the museum at time t = 0. (Your calculator should be in radian mode.)
Rate at which visitors enter the museum
| t (hours) | E(t) (people per hour) |
|---|---|
| 0 | 120 |
| 1 | 156 |
| 3 | 176 |
| 4 | 160 |
| 7 | 84 |
| 8 | 50 |
Source: Hypothetical data
Suggested time: 15 minutes
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Part (a)
3 pointsUse a trapezoidal sum with the five subintervals indicated by the table to approximate ∫₀⁸ E(t) dt. Show the computations that lead to your answer. Using correct units, explain the meaning of ∫₀⁸ E(t) dt in the context of the problem.
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Part (b)
2 pointsFind the average rate, in people per hour, at which visitors leave the museum over the time interval 0 ≤ t ≤ 8.
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Part (c)
2 pointsIs the number of visitors in the museum increasing or decreasing at time t = 4? Give a reason for your answer.
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Part (d)
2 pointsMust there be a time t, for 3 < t < 7, at which E(t) = L(t)? Justify your answer.
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