   Chapter 3.7, Problem 75E

Chapter
Section
Textbook Problem

# An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sight θ between the runners. [Hint: Maximize tan θ .] To determine

To find:

The maximum value of the observer’s angle of sight θ between the runners

Explanation

1) Concept:

If f'x<0 then fx is the absolute minimum value

Formula:

tanα+θ=tanα+tanθ1-tanα·tanθ

2) Calculation:

An observer stands at a point P, one unit away from the track

We have to find value of θ

tanα=Side opposite to αSide adjacent to α

=1t1

=t

tanα+θ=Side opposite to α+θSide adjacent to α+θ

tanα+θ=3t1

By using formula,

tanα+tanθ1-tanα·tanθ=3t

Substitute value of tanα

t+tanθ1-t·tanθ=3t

t+tanθ=3t1-t·tanθ

t+tanθ=3t-3t2·tanθ

tanθ+3t2·tanθ=3t-t

1+3t2

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