5
advanced-math

A gutter is made by turning up the edges of a 12-inch wide sheet of metal. What depth maximizes the cross-sectional area?

A

3 inches

B

4 inches

C

6 inches

D

2 inches

Correct Answer: A

Choice A is the correct answer.

  1. Setup: Let xx be the depth (height of turned-up sides). The base width is 12−2x12 - 2x.
  2. Area Function: A(x)=x(12−2x)=−2x2+12xA(x) = x(12 - 2x) = -2x^2 + 12x.
  3. Maximize: Vertex x=−12/(2(−2))=3x = -12 / (2(-2)) = 3.
  4. Result: Depth is 3 inches.

Choice B is incorrect. Choice C is incorrect because width would be 0. Choice D is incorrect.

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