Free response without a calculator (Part B)
Rectangle under a parabola: related rates and optimization
- Units 3, 4 and 5
- 9 points
- About 15 minutes
A multi-part problem you solve by hand, often from a graph, a table, an equation or a differential equation. You show your work, use exact values, and justify answers with calculus reasons such as a sign change in a derivative or the conditions of a theorem. On the exam: 4 questions in Part B of the free-response section (60 minutes, no calculator). 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, Part B usually includes a series question and a question on BC-only topics.
The question
A rectangle has its base on the x-axis and its two upper vertices on the graph of y = 12 − x². Its lower right vertex is at the point (x, 0), where 0 < x < 2√3, so the rectangle has width 2x and height 12 − x². The value of x changes with time t, measured in seconds. All lengths are in centimeters.
Suggested time: 15 minutes
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Part (a)
2 pointsAt the instant when x = 1, x is increasing at a rate of 1/2 centimeter per second. Find the rate at which the area of the rectangle is changing at that instant. Indicate units of measure.
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Part (b)
2 pointsAt the same instant (x = 1 and dx/dt = 1/2), find the rate at which the length of a diagonal of the rectangle is changing.
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Part (c)
3 pointsFind the greatest possible area of the rectangle. Justify your answer.
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Part (d)
2 pointsLet θ be the angle between the x-axis and the diagonal that runs from the point (−x, 0) to the point (x, 12 − x²). Find the rate at which θ is changing at the instant when x = 1 and dx/dt = 1/2.
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