Question Detail
Question
A child has 200ft of fence wire and wants to enclose a rectangular area for pet gerbils. Of all possible rectangular areas he could enclose, what is their average area (round to the nearest integer)?
Correct Answer
Option B is the correct answer.
Detailed Explanation
With perimeter 200, one side is x and the other is 100-x, so A(x)=x(100-x) for 0<x<100. The average value of A(x) over this interval is (1/100)∫₀¹⁰⁰ x(100-x)dx = 5000/3 ≈ 1667 ft².
Hint
Express area as a function of one side length, then find its average value over the valid range using integration.