Free response with a graphing calculator (Part A)
Grain pile: accumulation and related rates
- Units 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
Grain is poured onto a pile at a rate modeled by G(t) = 6 + 20t·e^(−t/5) cubic feet per minute, for 0 ≤ t ≤ 10, where t is measured in minutes. The pile is always shaped like a right circular cone whose base radius equals its height. At time t = 0, the pile contains 300 cubic feet of grain. (The volume of a cone with radius r and height h is V = (1/3)πr²h.)
Suggested time: 15 minutes
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Part (a)
2 pointsFind the average rate, in cubic feet per minute, at which grain is poured onto the pile over the time interval 0 ≤ t ≤ 10.
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Part (b)
2 pointsFind G′(7). Using correct units, interpret the meaning of G′(7) in the context of the problem.
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Part (c)
2 pointsFind the volume of the pile at time t = 10.
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Part (d)
3 pointsFind the rate at which the height of the pile is changing at time t = 10.
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