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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

An isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the point ( 0 , a ) on the y-axis (see the figure). Find the value of a that minimizes the lengths of the equal sides. (You may be surprised that the result does not give an equilateral triangle.).

Chapter 3.P, Problem 10P, An isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the

To determine

To find:

The value of a that minimizes the lengths of the equal sides.

Explanation

1) Concept:

Use implicit differentiation to unit circle to find the slope of AC- at point D then find the tangent line at D(b, c) to find the value of a that minimizes the lengths of the equal sides.

2) formula:

i. Power rule:ddxxn=nxn-1

ii. Quadratic formula: If ax2+bx+c=0 then x=-b±b2-4ac2a

iii. Quotient rule-ddxuv=v·ddxu-u·ddxvv2

3) Calculation:

Let AC- be tangent to the unit circle at point D.

To find the slope of AC- at point D.

Use implicit differentiation to unit circle.

x2+y2=1

Differentiate implicitly.

2x2-1+2y2-1y'=0

Simplify.

2x+2yy'=0

Subtract 2x and simplify.

2x+2yy'-2x=0-2x

2yy'=-2x

Divide by 2y on both sides.

2yy'2y =-2x2y 

Simplify.

y'=-xy =-bc

Therefore, the tangent line at D(b, c) has equation.y=-bc x+a

At D, x=b, y=c

So put x=b, y=c in the tangent line equation. y=-bc x+a

c=-bc (b)+a

Simplify and add b2c  on both sides.

c+b2c =-b2c +a+b2c 

Simplify and rearrange.

a=c2+b2c 

Since c2+b2=1

a=1c 

That is c=1a 

Since c2+b2=1

Subtract c2 from both the sides.

c2+b2-c2=1-c2

Simplify.

b2=1-c2

Take square root of both the sides.

b2=1-c2

Simplify.

b=1-c2

Put c=1a  in the above equation.

=1-1a 2

Simplify.

b=a2-1a

At C, y=-1

So put y=-1 in the tangent line equation. y=-bc x+a

-1=-bc x+a

Add bc x+1 on both sides.

-1+bc x+1 =-bc x+a+bc x+1

Simplify

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