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advanced-math

A farmer has 100 meters of fencing to enclose a rectangular area next to a river (no fence needed on the river side). What is the maximum area?

A

1250 sq meters

B

2500 sq meters

C

1000 sq meters

D

5000 sq meters

Correct Answer: A

Choice A is the correct answer.

  1. Setup: Let xx be the width (perpendicular to river). The length (parallel to river) is 100−2x100 - 2x.
  2. Area Function: A(x)=x(100−2x)=−2x2+100xA(x) = x(100 - 2x) = -2x^2 + 100x.
  3. Maximize: Find the vertex x=−b/2a=−100/(2(−2))=25x = -b/2a = -100 / (2(-2)) = 25.
  4. Calculate Area: A(25)=25(100−50)=25(50)=1250A(25) = 25(100 - 50) = 25(50) = 1250 sq meters.

Choice B is incorrect because it assumes a square with 4 sides (25×2525 \times 25 is wrong perimeter). Choice C is incorrect. Choice D is incorrect.

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Set 16: Quadratic Equations and Functions 16