Optimization problem: A sector of angle θ is cut from a circle of radius 9 inches and the resulting edges are brought together to form a cone. Find the magnitude of θ so that volume of the cone is maximized.

Question

anonymous · Answer

If you're also seeing a line of question marks... it's supposed to be $\theta$.

anonymous · Answer

Hint: you have two equivalent expressions for area ($A$) in terms of two different variables; one which you find in the volume equation ($r$), and the other is the variable you're optimizing ($\theta$).

You also have an expression that relates the other unknown in the volume eq ($h$) in terms of the radius ($r$), which in turn can be expressed in terms of the angle ($\theta$).

anonymous · Answer

All this means you can write volume in terms of the angle.

anonymous · Answer

yeah, Thanks! Had a hard time writing the volume in terms of angle.

anonymous · Answer

yw

anonymous · Answer

Oh hold up, I made a mistake somewhere along the line!

The $r$ you see above the circle is the arc length, but is not the same as the $r$ on the cone ($r$ is NOT the cone's radius). Let me draw that again with correct variables:
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