   Chapter 7.6, Problem 26E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Finding Distance In Exercises 23-28, find the minimum distance from the curve or surface to the given point. (Hint: Start by minimizing the square of the distance.)Circle:   ( x   −   4 ) 2 +   y 2 =   4 ,   ( 0 ,   10 ) Minimize d 2 =   x 2 + ( y   − 10 ) 2

To determine

To calculate: The minimum distance from cone (x+4)2+y2=4 to the point (0,10) minimize

d2=x2+(y10)2.

Explanation

Given Information:

The distance function is d2=x2+(y10)2 and the constant function is (x+4)2+y2=4.

Formula used:

If f(x,y) has a maximum or minimum subject to the constraint g(x,y)=0 then it will occur at one of the critical numbers of the function F defined by

Fx(x,y,λ)=0Fy(x,y,λ)=0Fλ(x,y,λ)=0

Where λ= Lagrange multiplier

Step 1: Solve the above system of equations.

Step 2: Evaluate f at each solution point obtained in the in the first step. The greatest value yields the maximum of f subject to the constraint g(x,y) and the least value yields the minimum of f subject to the constraint g(x,y)=0

Calculation:

Consider the given function,

For the minimum distance the function f is defined as

f(x,y,λ)=x2+(y10)2+λ[(x+4)2+y24]Fx(x,y,λ)=02x+2(x+4)λ=0λ=xx+4

And,

Fy(x,y,λ)=02(y10)+2yλ=0λ=(y10)y

And,

Fλ(x,y,λ)=0(x+4)2+y24=0 …… (1)

On simplify the equation,

5x2y

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