how to find the maximum volume of a cylinder within a sphere
"A circular cylinder is to fit inside a sphere of radius 10cm. Calculate the maximum possible volume of the cylinder. (It is probably best to take as your independant variable the height, or half the height, of the cylinder.)"
Cambridge A-levels pure mathematics chapter 7 (applications of differentiation), so differentiation will likely be part of the solution.
The radius of the sphere is of 10 cm.
The angle made by the radius of the sphere, with the edge of cylinder that intercepts the surface of the sphere is "x".
Now, we'll determine the radiusof cylinder in therms of x.
r cyl. = r*sin x
height of cyl. is h = 2r*cos x
Volume of cylinder = pi*(r cyl.)^2*h
Volume of cylinder = 2pi*r^3*(sin x)^2*(cos x)
We'll use Pythagorean identity:
Volume of cylinder = 2pi*r^3*[1 - (cos x)^2]*(cos x)
Volume of cylinder = 2pi*r^3*(cos x) - 2pi*r^3*(cos x)^3
To maximize the volume, we'll have to determine the first derivative of the function of volume. We'll differentiate with respect to x.
dV/dx = - 2pi*r^3*sin x + 6pi*r^3*(cos x)^2*sin x
We'll cancel dV/dx;
dV/dx = 2pi*r^3*sin x *(-1 + 3*(cos x)^2) = 0
2pi*r^3*sin x* (-1 + 3*(cos x)^2) = 0
We'll cancel each factor:
2pi*r^3*sin x = 0 => x = 0
(-1 + 3*(cos x)^2) = 0
3*(cos x)^2 = 1
cos x = + sqrt3/3 or cos x = - sqrt3/3
V = 2pi*1000*(sqrt3/3 - sqrt3/9)
Therefore, the maximum volume is:V = 2pi*1000*2sqrt3/9 cm^3