Question

A rectanglar plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for $6 a foo, while the remaining two sides will use the standard fencing selling at $4 a foot. What are the dimensions of the rectanglar plot of greatest area that can be fenced at a cost of $12000?

Answer 1

Dimensions of the rectangular plot are 500 x 750 ft

Explanation:

First we need to set-up the equation for the total cost of the fence. So, let's use a $6 per foot fencing for the two sides "x" . And, $4 per foot fencing for the two sides "y".

So the equation is:

12000 = 6(2x) + 4(2y)

12000 = 12x + 8y

To simplify, divide both sides by its GCF which is 4

3000 = 3x + 2y

Solving for y.

3000 - 3x = 2y

y = 1500 - 3/2x

To maximize the area of the rectangle, set-up the equation of area.

A = xy

Substitute value of y,

A = x(1500-3/2x)

A = 1500x - 3/2x^2

Take derivative of A

A' = 1500 - 3x

Set A' equal to zero and solve for x.

0 = 1500 - 3x

3x = 1500

x = 500

Substitute the value of x = 500 to y = 1500 - 3/2x

y = 1500 - 3/2(500)

y = 1500 - 750

y = 750

Hence, the dimensions of the rectangular plot that would maximize its area, given the total cost of fencing 12000, is 500 x 750 ft.

Answer 2

250 x 1125ft

Explanation:

Let x and y be the sides of the rectangle.

Then, set-up the equation for the total cost of the fence.

To do so, given that $6 per foot fencing for the two sides "x" . And, $4 per foot fencing for the two sides "y".

So the equation is

12000 = 6(2x) + 4 (2y)

12000 = 12x + 8y

To simply, divide both sides by its GCF which is 2

12000/2 = (12x + 8y)/2

6000 = 6x + 4y

Then solve for y

6000-6x = 4y

=> (6000-6x)/4 = y

=> 1500 - (6/4)x = y

Since we have to maximize the area of the rectangle, set up the equation of the area.

Area = xy

Note that y = 1500 - (6/4)x

Therefore, A = x (1500 -6/4x)

A=1500x - 6/4x^2

Then, take the derivative of A.

A' =1500 - 6x

Set A' equal to zero and solve for x.

0 = 1500-6x

6x = 1500

x = 250

Substitute the value of x to y = 1500 - 6/4x .

y = 1500 - 6/4 * 250 = 1500 - 375 = 1125

Hence, the dimensions of the rectangular plot that would maximize its area, given the total cost of fencing, is 250 x 1125 ft.