"Published in Vacaville, California, USA"
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?
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| Photo by Math Principles in Everyday Life |
Solution:
From the given figure, it is a cross section of a gutter in a form of isosceles trapezoid because the opposite sides as well as the opposite angles of the lower base are congruent.
In the given word problem, maximum amount means maximum volume. Since the given figure is a cross section and the height of a gutter is not given, then it is a maximum area.
If the length of the opposite sides of a trapezoid as well as the opposite angles at the lower base are given, then we can solve for the altitude and the length of the upper base.
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| Photo by Math Principles in Everyday Life |
Since the two opposite triangles at the opposite sides of a trapezoid are right triangles, then we can use the trigonometric functions as follows
The area of a cross section of a gutter which is an isosceles trapezoid is
Take the derivative on both sides of the equation with respect to θ, we have
but
then the above equation becomes
Equate each factors to zero and solve for the value of θ.
Since the value of θ is a straight angle, then we cannot accept this as an answer. Let's consider the other factor and solve for the value of θ, we have
Since the value of θ is an acute angle, then we can consider this one as an answer.