# Math Help - A bio calculus question

1. ## A bio calculus question

For this problem I got as far as to getting the general equation of V(t)=(Area of Trapezoid)*Length. I'm not sure how to write out the Area part of the equation. If anyone can, please help me on this problem. Thanks!

This problem examines two classic problems in optimization that use trigonometric functions. In both problems we use the same amount of material to construct a feeding trough (not including the ends). The feeding troughs are made by bending a 100 cm long strip of sheet metal that is 45 cm wide. The metal sheet is bent so that the 45 cm width creates a trapezoid with the trough maintaining a length of 100 cm.

a. For the first trough the sheet metal is bent into an isosceles trapezoid by turning up strips that are x = 15 cm wide on each side so that they make the same angle, , relative to a line perpendicular to the bottom.

(see example)
http://www-rohan.sdsu.edu/~jmahaffy/.../labd/q1v1.htm

Write a function of the volume depending on . (use t for )

V(t)=__________________________

2. Finding an expression $A(\theta)$ for the area $A$ of the trapezoid as a function of $\theta$ the angle of the flaps to the vertical, is the real meat of the problem.

First, remember that $A=\frac12(b_1+b_2)h$ for an isosceles trapezoid with top and bottom bases $b_1,b_2$, and height $h$. According to the picture, $b_1=15, h=15\cos\theta, b_2=15+2*15\sin\theta$, so the area is $A(\theta)=15^2(1+\sin\theta)\cos\theta$

I think you can take it from there.

3. Thank you so much for the help!