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Chapter 3: Q. No. 7 (page 165)

Enclosing a Rectangular Field: David has 400 yards of fencing and wishes to enclose a rectangular area.

(a) Express the area A of the rectangle as a function of the width w of the rectangle.

(b) For what value of w is the area largest?

(c) What is the maximum area?

Short Answer

Expert verified

(a) The area A of the rectangle as a function of the width w of the rectangle is A=-w2+200w.

(b) The area is the largest when w=100.

(c) The maximum area is 10,000 square yards.

Step by step solution

01

Part (a) Step 1. Given information.

Given that David has 400 yards of fencing and wishes to enclose a rectangular area.

02

Part (a) Step 2. The area as a function of width of the rectangle.

The perimeter of the rectangular area is 400yards.

Let the width of the rectangle be w which means

2(l+w)=400l+w=200l=200-w

The area A as a function of the width of the rectangle is

A=lwA=(200-w)wA=200w-w2A=-w2+200w

03

Part (b) Step 1. The value of w for the largest area.

The function A is a quadratic function with a=-1,b=200, and c=0. Because a<0, the vertex is the highest point on the parabola.

The areaAis a maximum when the width w is

w=-b2a=-2002(-1)=100

04

Part (c) Step 1. Substitute x=100 in A=-w2+200w.

We get

A=-(100)2+200(100)=-10000+20000=10000

The maximum area is 10,000 square yards

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