I want to work out the equation for volume of half a sphere against the surface area of the circle at the widest part of the sphere.
The equation of the half sphere is (2/3) * (pi * r ^3), where r is the radius, and obviously the area of the circle is pi *(r ^2).
Basically I want a function like: surface area = (something) * (volume ^ something) + something.
Any hints would be great. Thanks!
The volume of a hemisphere is $V=\frac 23 \pi r^3$ and as you say the area of a circle is $A=\pi r^2$ You can't have a direct proportion between them as the units do not match. As $r$ increases, the volume increases faster than the area. What you can have is $V=kA^{3/2}$ or $A=k'V^{2/3}$ If you plug in the formulas for $V,A$ you can evaluate $k,k'$.