Question

A gardener wants to design a rectangular garden and has 100 feet of fencing available. A house is on one side of the garden, so no fencing will be needed there. What dimensions will give the maximum area?

Answer 1

• 50 ft and 25 ft give the maximum area

Explanation:

Let l and w be the dimensions of rectangle.

The length of fencing is 100 ft:

• l + 2w = 100

Then:

• l = 100 - 2w

The area is:

• A = lw = w(100 - 2w) = 100w - 2w²

In order to have the maximum area we need to find the vertex of the quadratic function:

• y = - 2w² + 100w

The vertex has x-coordinate:

• w = - b/(2a) It comes from quadratic function y = ax² + bx + c

or

• w = - 100/(2*(-2)) = 25

So the width is 25 ft.

Find the length:

• l = 100 - 2*25 = 50 ft

The area is:

• A = 50*25 = 1250 ft²

See the attached graph.