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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. II the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon.

images

To determine

To find: The lengths of the sides of a pentagon which maximize the area of a pentagon.

Explanation

Given:

Notice that the given pentagon has one rectangle and one isosceles triangle.

Let a and c be the side lengths of the isosceles triangle and the angle between a and c is θ . Let, b and c be the length, breadth of a rectangle.

Thus, the perimeter of the pentagon is fixed.

The above information mentioned in the following Figure 1.

Result used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx , fy=λgy , fz=λgz and g(x,y,z)=k ”.

Calculation:

The area of the triangle is, 12casinθ and the area of a rectangle is bc.

Thus, the area of the pentagon is, f(a,b,c)=12casinθ+bc .

The perimeter of the pentagon is, P=g(a,b,c)=2a+2b+c .

To obtain the value of sinθ in terms of a, b, c, use Pythagoras theorem,

From the triangle ABD, AC=asinθ .

From triangle ABD,

AB2=AD2+BD2a2=(c2)2+(asinθ)24a2c24=a2sin2θsin2θ=4a2c24a2

Simplify further as follows.

sinθ=4a2c24a2=12a4a2c2

Substitute sinθ=12a4a2c2 in f(a,b,c)=12casinθ+bc ,

f(a,b,c)=12ca(12a4a2c2)+bc=14c4a2c2+bc

By above result f(a,b,c)=λg(a,b,c) is computed as follows.

f(a,b,c)=λg(a,b,c)fa,fb,fc=λga,gb,gcfa(14c4a2c2+bc),fb(14c4a2c2+bc),fc(14c4a2c2+bc)=λga(2a+2b+c),gb(2a+2b+c),gc(2a+2b+c)ac4a2c2,c,(144a2c2c244a2c2+b)=λ2,2,1

Thus, the value of f(a,b,c)=λg(a,b,c) is, ac4a2c2,c,(144a2c2c244a2c2+b)=λ2,2,1

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Chapter 14 Solutions

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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