# The sum | P A | + | P B | + | P C | is a minimum.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4, Problem 43RE
To determine

## To find:The sum |PA|+|PB|+|PC| is a minimum.

Expert Solution

PD2.31, PA||CD

### Explanation of Solution

Given:

PA=PB=4secθ

PC=CDPDPC=54tanθ

Concept used:

Triangle ABC with altitude CD

PC=CDPD

Calculation:

Triangle ABC with altitude CD

PA=PB=4secθ.......................(1)

PC=CDPDPC=54tanθ

Differentiating equation (1)

dSdθ=8.secθ.tanθ4sec2θdSdθ=08.secθ.tanθ4sec2θ=04.secθ(2tanθsecθ)=0

secθ02tanθ=secθ2sinθcosθ=1cosθcosθ02sinθ=1sinθ=12

Therefore

θ=π6=300

P must be placed so that

PD=4.tan300PD=4.13PD43PD2.31

If cosθ=0

Then θ=900

And PA||CD

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