What dimensions would guarantee that the garden has the greatest possible area?

Question

A gardener has 920 feet of fencing to fence in a rectangular garden. One side of the garden is bordered by a river and so it does not need any fencing.

Answer 1

Well, call the width X and the length (920 - 2x). So the area is X (920 - 2x).. We can graph this using our graphing calculator and the result is a parabola that opens down with its max at about x = 230. Alternatively, we can take the derivative, set it equal to zero and get the same result, as show here.

d/dx of - 2x^2 - 920 x is - 4x + 920. Set it equal to zero and solve for x and you get 230 again.

Answer 2

We always begin by drawing a diagram. The diagram helps us to visualize the problem. We then interpret it to an equation. In this case the equation is the length of the field plus the two widths must equal our available fencing.

To solve we must isolate one variable and input that into the formula we desire to maximize (in this case the area of the field). See the video for a step by step procedure.