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 doe

Question

1 year ago

10

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 doe

s 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)

Mathematics

1 answer:

Bond [772] · Answer 1 · 2023-11-19T11:09:57+03:00

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

<h3>What dimensions would guarantee that the garden has the greatest possible area?</h3>

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