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Chapter 3: Q. No. 9 (page 165)

Enclosing the Most Area with a Fence: A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed? ( See the figure)

Short Answer

Expert verified

The maximum are enclosed here is2,000,000square meters.

Step by step solution

01

Step 1. Given information.

Given that a farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river and the farmer does not fence the side along the river,

02

Step 2. Express the area A as a function of x (length of the rectangle).

Let the length of the rectangular field be x and

the width of the rectangle is 4000-2x.

Because 2l+w=40002x+w=4000w=4000-2x

The area of a rectangle is A=lw.

We get

A=x(4000-2x)=4000x-2x2=-2x2+4000x

03

Step 3. Find the maximum area enclosed.

The function A is a quadratic function with a=-2,b=4000, and c=0. As a<0, the vertex is the highest point on the parabola.

The area A is maximum when x is

x=-b2a=-40002(-2)=1000

Substitute x=1000in A=-2x2+4000x.

A=-2(1000)2+4000(1000)=-2000000+4000000=2,000,000

The maximum area enclosed is 2,000,000 square meters.

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