Height of a spherical cap with known volume?

[streeters]

Homework Statement

What equation is needed to calculate the height of a spherical cap with a fixed volume and radius?

Homework Equations

V=πh/6 (3a^2 + h^2)
Where V = volume, h is cap height, a is cap radius

The Attempt at a Solution

I have tried to separate the h out and got as far as:
6V/π = h[(√3a+h)^2 -2√3ah]

but think I am on the wrong track

 

[SteamKing]

Homework Statement

What equation is needed to calculate the height of a spherical cap with a fixed volume and radius?

Homework Equations

V=πh/6 (3a^2 + h^2)
Where V = volume, h is cap height, a is cap radius

The Attempt at a Solution

I have tried to separate the h out and got as far as:
6V/π = h[(√3a+h)^2 -2√3ah]

but think I am on the wrong track

An alternate formula for the volume of a spherical cap can be found here:

http://en.wikipedia.org/wiki/Spherical_cap

V = (πh²/3)*(3r - h), where h is the height of the cap and r is the radius of the sphere.

 

[Ray Vickson]

Homework Statement

What equation is needed to calculate the height of a spherical cap with a fixed volume and radius?

Homework Equations

V=πh/6 (3a^2 + h^2)
Where V = volume, h is cap height, a is cap radius

The Attempt at a Solution

I have tried to separate the h out and got as far as:
6V/π = h[(√3a+h)^2 -2√3ah]

but think I am on the wrong track

Your equation for ##h## is a cubic. There are formulas for solutions of cubic equations, so you can get ##h## in terms of ##a## and ##V##. In fact, you have a so-called "depressed cubic" where ##h## appears only in ##h^3## and ##h## terms (with no ##h^2## term), and for such equations the solution is a bit simpler. See, eg., http://en.wikipedia.org/wiki/Cubic_function or
http://www.sosmath.com/algebra/factor/fac11/fac11.html .