Aim for FiveCalculus AB

Free response with a graphing calculator (Part A)

Water in a storage tank

  • Units 4, 5, 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

Water is pumped into a storage tank at a rate modeled by I(t) = 80 + 40 sin(t²/16) liters per hour, for 0 ≤ t ≤ 8, where t is measured in hours. During the same time interval, water drains out of the tank at a rate modeled by O(t) = 90 + 15 cos(t/2) liters per hour. At time t = 0, the tank contains 1200 liters of water. (Your calculator should be in radian mode.)

Suggested time: 15 minutes

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Part (a)

2 points

Find I′(6). Using correct units, interpret the meaning of I′(6) in the context of the problem.

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Part (b)

2 points

Find the total amount of water pumped into the tank from time t = 0 to time t = 8. Show the setup for your calculations.

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Part (c)

3 points

Let W(t) be the amount of water in the tank, in liters, at time t. At what time t, for 0 ≤ t ≤ 8, is the amount of water in the tank at an absolute minimum? Justify your answer.

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Part (d)

2 points

For t > 8, no water is pumped into the tank, and water continues to drain out at the rate O(t) liters per hour. Write, but do not solve, an equation involving an integral expression that can be used to find the time t = k when the tank first becomes empty.

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