   Chapter 4.7, Problem 65E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.

To determine

To prove: All isosceles triangles with a perimeter, the one with the greatest area is equilateral.

Explanation

Formula used:

Pythagorean Theorem:

If c denotes the length of the hypotenuse and a and b denote the length of the other two sides of a right angle triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation: a2+b2=c2

Proof:

Consider a triangle as below.

In Figure 1, an isosceles triangle whose base is b and the sides are s.

s2=h2+b24h2=s2b24

Thus, the area is A=12bs2b24.

Let the perimeter be p.

2s+b=ps=pb2

Substitute the value of s=pb2 in A=12bs2b24

A=12b(pb2)2b24=12b.p22pb+b24b24=14bp22pb

Differentiate A with respect to b,

For critical points,

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