equation on finding the height for a given volume on a sphere?

Question

The only thing im given is that we find the volume by integrating

v(H)= ∫ πr²dh..from o to H, where r²=R²-(R-h)²

I evaluated V(H) which gave me

V(H)=πRH²-((πH²)/3), so I figured if that gave me the volume I should just solve for H and find the equation but I havent been able to solve for H. Can anyone help me please.

Answer 1

Your question is not very clear. I think you are asking for the volume of a spherical cap. For example, if you had a hemispherical bowl with radius R, filled with water to a depth h, what is the volume of water in the bowl?

Integrating over y from 0 to h we have:

V = ∫π(√(R² - y²))² dy = ∫π(R² - y²)dy

V = π(R²y - y³/3) | [Evaluated from 0 to h]

V = π(R²h - h³/3) = (⅓πh)(3R² - h²)

Notice that when h = R the volume is 2πR³/3, which is the volume of a hemisphere as we would expect.

Here is a link to MathWorld regarding spherical caps.

http://mathworld.wolfram.com/SphericalCap.html

Answer 2

Volume of a sphere is 4/3 (pi) r^3
so, if the volume is 32/3 pi, height is 2
2^3 *4/3= 8*4/3 =32/3

Answer 3

well you could always just work backwards from the volume equation.

V = 4/3(pi)r^2
3V / 4(pi) = r^2
(squareroot) 3V / 4(pi) = r
2 (squareroot) 3V / 4(pi) = 2r = H