You have 800 ft of fencing to make a pen for hogs. if you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area?

As per the perimeter of a rectangle, the length and width of the rectangular pen that maximizes the area is 400 feet and 200 feet respectively.

What is the perimeter of a rectangle?

The perimeter of a rectangle is the sum of all four sides of the rectangle.

Given, I have 800 feet of fencing to make a pen for hogs.

Let, 'l' is the length and 'w' is the width of the rectangle.

I have a river on one side of your property.

Therefore, fencing on the three sides of my property is enough.

Therefore, required perimeter of the fencing [tex]p = l + 2w[/tex]

Now, perimeter of the fencing is equal to

[tex]l + 2w = 800[/tex]

⇒ [tex]l = 800 - 2w[/tex]

Now, the area of the rectangle [tex]A = lw[/tex]

Therefore, [tex]A = lw = w(800-2w) = 800w - 2w^{2}[/tex]

We can calculate the maximum area of the rectangle by differentiating the equation of the area.

Therefore, [tex]A' = 800-4w[/tex]

Again, [tex]A'' = -4[/tex]

As the second order differentiation is constant negative, therefore the area of the rectangle is maximum.

Now, putting [tex]A' = 0[/tex], at [tex]A' = 800-4w[/tex], we get:

[tex]800-4w = 0[/tex]

⇒ [tex]4w = 800[/tex]

⇒ [tex]w = \frac{800}{4}[/tex]

⇒ [tex]w = 200[/tex]

Therefore, width of the rectangle is 200 feet.

Now, the length of the rectangle is:

[tex]l = 800 - 2w = [800 - 2(200)] feet = 400 feet[/tex]

Learn more about the perimeter of a rectangle here: https://brainly.com/question/17096963

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