   Chapter 3.7, Problem 76E

Chapter
Section
Textbook Problem

# A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle θ . How should θ be chosen so that the gutter will carry the maximum amount of water? To determine

To find:

To choose the value of θ so that the gutter carries the maximum amount of water

Explanation

1) Concept:

First Derivative Test for Absolute Extreme Values:

Suppose that c is a critical number of a continuous function f defined on an interval.

(a) I f'x>0 for all x<c and f'x<0 for all x>c then fc is the absolute maximum value of f.

(b) If  f'x<0, 0 for all x<c and f'x>0 for all x>c then fc is the absolute minimum value of f.

2) Calculation:

From the figure,

sinθ=h10, cosθ=d10

h=10sinθ & d=10cosθ

So the amount of water is a function of θ

Aθ=10h+212dh

Substitute the values of d & h

Aθ=10·10sinθ+10cosθ·10sinθ

Aθ=100sinθ+cosθsinθ

For 0θπ2

Differentiate with respect to θ

A'(θ)=100(cosθ-sin2θ+cos2θ)

A'(θ)=100(cosθ-1+cos2θ+cos2θ)

A'θ=100cos2;

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