Arancher has 800 feet of fencing to put around a rectangular field and then subdivide the field into 2 identical smaller rectangular plots by placing a fence parallel to one of the field's shorter sides. find the dimensions that maximize the enclosed area. write your answers as fractions reduced to lowest terms.

The area of a ranch is the amount of space on the ranch.

The dimension that maximizes the enclosed area is: 400/3 by 200

The perimeter is given as: 800

This means that:

Where: x and y are the dimensions of the ranch

So, we have:

Make y the subject

The area of the ranch is:

Substitute

Split fraction

Differentiate

Set to 0

Collect like terms

Divide through by 3

Recall that:

So, we have:

 and

Hence, the dimension that maximizes the enclosed area is: 400/3 by 200

Read more about areas at:

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The dimensions of enclosed area are 200 and 400/3 feet

Step-by-step explanation:

* Lets explain how to solve the problem

- There are 800 feet of fencing

- We will but it around a rectangular field

- We will divided the field into 2 identical smaller rectangular plots

 by placing a fence parallel to one of the field's shorter sides

- Assume that the long side of the rectangular field is a and the

 shorter side is b

∵ The length of the fence is the perimeter of the field

∵ We will fence 2 longer sides and 3 shorter sides

∴ 2a + 3b = 800

- Lets find b in terms of a

∵ 2a + 3b = 800 ⇒ subtract 2a from both sides

∴ 3b = 800 - 2a ⇒ divide both sides by 3

∴ ⇒ (1)

- Lets find the area of the field

∵ The area of the rectangle = length × width

∴ A = a × b

∴

- To find the dimensions of maximum area differentiate the area with

  respect to a and equate it by 0

∴

∵

∴ ⇒ Add 4/3 a to both sides

∴ ⇒ multiply both sides by 3

∴ 800 = 4a ⇒ divide both sides by 4

∴ 200 = a

- Substitute the value of a in equation (1)

∴

* The dimensions of enclosed area are 200 and 400/3 feet

step-by-step explanation: