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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.

To determine

To find: The dimension of the isosceles triangle with the largest area, inscribed inside a circle of radius r.

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

Calculation:

Let an isosceles triangle is inscribed inside a circle with a radius r.

Consider Figure 1, the base of the triangle is b and the height of the triangle is (r+x) .

Then, by Pythagorean Theorem,

x2+(b2)2=r2(b2)2=r2x2b=2(r2x2)

The area of the triangle is,

A=12×(Base)×(Height)=12b(r+x)

Substitute b=2(r2x2) in A=12b(r+x) ,

A=12b(r+x)=12(2(r2x2))(r+x)=(r2x2)(r+x)

Differentiate A with respect to x,

dAdx=(r2x2)ddx(r+x)+(r+x)ddx((r2x2))=(r2x2)+(r+x)×122x(r2x2)=(r2x2)(x2+rx)(r2x2)

For critical points, dAdx=0 ,

(r2x2)(x2+rx)(r2x2)=0r2x2=x2+rx2x2+rxr2=0

Use quadratic formula,

x=r±r24(2)(r2)2(2)=r±9r24=r±3r4

x=r3r4=4r4=r

x=r+3r4=2r4=r2

x cannot be negative, so x=r2

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