Question

A gardener has 520 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.

garden bordered by a river

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

shorter side: _____ft (feet)

longer side: ____ft (feet)

greatest possible area: ___ft2 (square-feet)

1. mocmien2
The shortest side is 130 feet, the longest side is 260 feet and the greatest possible area is 33800 square feet

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

The given parameter is
Perimeter, P = 520 feet
Represent the shorter side with x and the longer side with y
One side of the garden is bordered by a river:
So the perimeter is:
P = 2x + y
Substitute P = 520
2x + y = 520
Make y the subject
y = 520 – 2x
The area is
A = xy
Substitute y = 520 – 2x in A = xy
A = x(520 – 2x)
Expand
A = 520x – 2x^2
Differentiate
A’ = 520 – 4x
Set to 0
520 – 4x = 0
Rewrite as:
4x= 520
Divide by 4
x= 130
Substitute x= 130 in y = 520 – 2x
y = 520 – 2 *130
Evaluate
y = 260
The area is then calculated as:
A = xy
This gives
A = 130 * 260
Evaluate
A = 33800
Hence, the shortest side is 130 feet, the longest side is 260 feet and the greatest possible area is 33800 square feet