## A rancher has 800 feet of fencing to put around a rectangular field and then subdivide the field into 2 identical smaller rectangular plots

Question

A rancher 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.

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1 day 2021-09-15T18:59:27+00:00 1 Answer 0

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