Question

A man wants to build a rectangular enclosure for his herd. He only has $900 to spend on the fence and wants to maximize the grazing area for his money. He plans to build the pen along the river on his property, so he does not have to put a fence on that side. The side of the fence parallel to the river will cost $5 per foot to build, whereas each side perpendicular to the river will cost $3 per foot. What dimensions should he choose?

Answer 1

The side of the fence parallel to the river : 90

The side perpendicular to the river: 75

Explanation:

Let x is the side of the fence parallel to the river

Let y is the side perpendicular to the river

Given that:

• He only has $900
• The side of the fence parallel to the river will cost $5
• Each side perpendicular to the river will cost $3 per foot

=> we have the equation of the total cost

5x + 2*3y = 900

<=> 5x + 6y = 900

<=> x = (900 - 6y)/5 (1)

We know that, the area of a rectangular enclosure for his herd is:

A = x*y (2)

Substitute (1) into (2) we have:

A = (900 - 6y)/5 * y

<=> A = 1/5 (900y - 6
`y^(2)` )

<=> A = -6/5
`y^(2)` + 180y

To find the maximum value of this, simply take the 1st derivative, and set it to 0

A' = 180 - 12/5y = 0

<=> y = 75

=> x = 90

So the dimensions he should choose is:

The side of the fence parallel to the river : 90

The side perpendicular to the river: 75