related rates question!

[sarahnguyen]

1. A water diversion channel is constructed by bending a sheet of metal that is 6 m wide, as shown below. a)What angle, ?, will maximize the cross-sectional area (and thus the capacity) of the channel?
b) What is this maximum area (express as an exact value!)

The picture wont work but its bent on one side at 2m, and the middle is 2m then the side is 2m as well. the angle lies between the ground and the bends of the metal.

 

[galactus]

This is not a related rates problem. It is an optimization problem.

Try thinking about a trapezoid. That is the shape after the sides are bent up.

The area of the trapezoid is \(\displaystyle A=\frac{1}{2}h(2+b)\)

\(\displaystyle h=2sin\theta, \;\ b=2+2(2cos\theta)\)

So, \(\displaystyle A=4sin\theta (cos\theta +1)\)

This is what is to be maximized.

 

[sarahnguyen]

thanks so much for jumpstarting me! i will come back to this if i get stuck thank you again