Finding Dimensions of Cone with Surface Area 1 and Max Volume

[skateza]

Homework Statement

The volume of a right circular cone is V = [(pie)(r^2)(h)]/3 and it ssurface area is S = (pie)(r)(r^2+h^2)^(1/2), where r is the base radius and h is the height of the cone. Find the dimensions of the cone with surface area 1 and maximum volume.

The Attempt at a Solution

I think the only difficult part of this question is the math, because its quite difficult. I'm finding V' to be
$$\pi r[r+(4/\pi^2r^2)-4r^2]/6[(1/\pi^2r^2)-r^2]$$
Which i can't find any zero's for, can someone double check this?

Steps to finding the derivative:,
1) Set S equal to 1 and solve for h,
2) stuff h into volume and take derivate, unless you know of a better way?

 

[Avodyne]

Your method is correct, but I get something simpler. After solving for h and plugging into V, simplify as much as possible before taking the derivative.

 

[skateza]

is this what you simplified it down to:
$$\sqrt{(1/9)r^2[1-\pi^2r^4]}$$.

If so, i got as a derivative:
$$(1/2)[(1/9)r^2(1-\pi^2r^4)]^(-1/2) [(1/9)r^2(-4\pi r^3) + (1-\pi^2r^4)(2/9)\pi]$$
which doesn't simplify down much nicer...

 

[Avodyne]

You're missing an r in the last term, but that's it. You want to set this to zero, so you can cancel out everything not in square brackets, and then pull out common factors and cancel those too ...