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1.9.26 Area between two curves (A-level only)

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Question 15

The rate of a chemical reaction, rrr grams per minute, is modelled by the function

r=9t−t1.5,t≥0 r = 9t - t^{1.5}, \quad t \ge 0 r=9t−t1.5,t≥0

where ttt is the time in minutes since the reaction began. A plot of this rate shows a curve starting at the origin, reaching a peak production rate at the point MMM, and then falling back to zero at the point PPP.

a.

Using calculus, find the coordinates of the maximum point MMM.

[4]
b.

The reaction concludes when the rate returns to zero at point PPP. Use algebra to find the ttt-coordinate of PPP.

[2]
c.

The finite region RRR is bounded by the curve, the horizontal line passing through MMM, and the vertical line passing through PPP.

Use algebraic integration to find the area of RRR, giving your answer to two decimal places.

[5]

1.9.26 Area between two curves (A-level only) Questions

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