calculus

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a trapezoid is a quadrilateral with (only) two sides (called bases) that are parallel. an isosceles trapezoid has two equal legs. the area of a trapezoid is

A=(1/2)*h*(b1+b2) where b1 and b2 are the two parallel bases and h is the altitude.

your problem: a water viaduct has a cross section that is an isosceles trapezoid. the area of the trapezoid varies as angle theta changes. you want to determine the angle theta that will maximize the amount of water this viaduct will conduct.

attempt at what it looks like:

\ | /
\81 h 81/
\ | /
(theta\___|_____/theta)
8

first determine a trigonometric expression for the height-find the height in terms of angle theta.

find an expression for the length of the upper base in terms of angle theta.

find an expression for the area of the viaduct in terms of angle theta.

use 3rd grade alebra to simplify the expression for the area-i.e. combine terms; factor our common factors, etc.

now use calculus to find the angle that will maximize the area of the viaduct, and thus maximize the volume of water flowing through it.

• calculus -

The diagram didn't make it. I can figure that theta is the angle the side makes with the horizontal, but where does the 81 come in?

• calculus -

dang it. well the diagram was what my teacher gave me to go by. the 81 is the diagonals on both sides, or that's the way its written on the sheet of paper. and 8 is the length of the base.

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