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...

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