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TMUA 2022 · Paper 1 · Question 15 of 20

TMUA 2022 Paper 1 Question 15

Differentiation and integration — Optimisation · rectangle between two curves. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Differentiation and integrationOptimisation · rectangle between two curves8 optionshard
A rectangle is drawn in the region enclosed by the curves p and q, where p(x) = 8 − 2x2 q(x) = x2 − 2 such that the sides of the rectangle are parallel to the x- and y-axes.

What is the maximum possible area of the rectangle?

  1. A269
  2. B529
  3. C4√63
  4. D8√63
  5. E4√2
  6. F8√2
  7. G20√109
  8. H40√109
Show the answer and worked solution
answer · H
  1. A269
  2. B529
  3. C4√63
  4. D8√63
  5. E4√2
  6. F8√2
  7. G20√109
  8. H40√109
Both curves are even, so the largest rectangle is symmetric about the y-axis, with vertical sides at x = ± w. Its height is limited at those edges, where the gap between the curves is (8 − 2w2) − (w2 − 2) = 10 − 3w2. So the area is A = 2w(10 − 3w2) = 20w − 6w3. Setting dAdw = 20 − 18w2 = 0 gives w = √103, and then A = 2√103⋅203 = 40√109.