   Chapter 3.7, Problem 74E

Chapter
Section
Textbook Problem

# A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the comer? To determine

To choose:

The length of longest pipe that can be carried horizontally around the corner

Explanation

1) Concept:

If f''x>0 if and only if fx is the absolute minimum value where f'x=0

2) Calculation:

Assume that L is the length of the pipe

Convert the length of pipe into two different pipes as L1 & L2

So, L=L1+L2

There are two angles equal to θ which are corresponding angles of the line intersecting two parallel lines

In ABC & CDE,

By using the formulae of trigonometric ratios of a triangle

sinθ=9L1, cosθ=6L2

Find out L1 & L2

L1=9sinθ, L2=6cosθ

L1=9cscθ, L2=6secθ

So, substitute these values in equation of L

L=9cscθ+6secθ

Differentiate with respect to θ

dLdθ=-9cscθcotθ+6secθtanθ

To find the minimum of LdLdθ=0

To find the value of θ

0=-9cscθcotθ+6secθtanθ

9cscθcotθ=6secθtanθ

secθtanθcscθcotθ=96

By using trigonometric formulae,

sin3θcos3θ=32

tan3θ=32

tanθ=313213

tanθ= opposite sides divided by the adjacent side,

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