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

Answers ( )

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    2021-09-15T19:01:09+00:00

    Answer:

    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

    b=\frac{800}{3}-\frac{2a}{3} ⇒ (1)

    – Lets find the area of the field

    ∵ The area of the rectangle = length × width

    ∴ A = a × b

    A=(a).(\frac{800}{3}-\frac{2a}{3})=\frac{800a}{3}-\frac{2a^{2}}{3}

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

      respect to a and equate it by 0

    \frac{dA}{da}=\frac{800}{3}-\frac{4a}{3}

    \frac{dA}{da}=0

    \frac{800}{3}-\frac{4}{3}a=0 ⇒ Add 4/3 a to both sides

    \frac{800}{3}=\frac{4}{3}a ⇒ multiply both sides by 3

    ∴ 800 = 4a ⇒ divide both sides by 4

    ∴ 200 = a

    – Substitute the value of a in equation (1)

    b=\frac{800}{3}-\frac{2}{3}(200)=\frac{800}{3}-\frac{400}{3}=\frac{400}{3}

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

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