Question

Hal has a 20-foot piece of wood to make a sandbox. What is the most efficient use of the

lumber to get maximum space with his timber? What is the maximum area of the sandbox
that can be created?

Answer 1

Answer: Build a 5 ft by 5 ft square, of area 25 sq ft

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Work Shown:

L = length

W = width

P = perimeter of a rectangle = 20

2L+2W = P

L = (P-2W)/2

Let x = W, so L = (20-2x)/2 = 10-x

A = area

A = length*width

A = (10-x)*x

A = -x^2+10x

The goal is to max out the function f(x) = -2x^2+10x. Plot this on a graph to see a parabola forms. The highest point is at (5,25) which is the vertex.

Therefore, the max area is 25 square feet and occurs when x = 5 is the width and 10-x = 10-5 = 5 is the length.

So he should form a 5 by 5 square of area 5*5 = 25

Notes:

• Perimeter of a square = 4*(side length)
• side length of a square = (perimeter of a square)/4