## Calculus Tutorial

#### Intro

Find the minimal volume and dimensions of a right circular cone circumscribed about a sphere of a given volume. To solve this problem we need to know

1) The formula for the volume of a sphere

2) The formula for the volume of a cone

3) The radius of a sphere is perpendicular to a tangent line to the sphere.

4) Setting up ratios using similar triangles.

#### Sample Problem

Find the volume, radius, and height of a right circular cone of smallest volume about a sphere of volume .

#### Solution

Let represent the volume of the sphere, x represent the radius of the sphere and , r and h represent the volume, radius and height of the right circular cone. I will solve the problem in general for any volume of a sphere.

1) Determine the radius of the sphere.

2) Set up similar triangles using the geometry of the problem and represent the volume of a cone a function of h.

Squaring each side gives

Now solve for in terms of h.

, cross multiplying

, move all terms containing to the right

, factor out

, foil, combine like terms then divide each side by

Now substitute this value for into formula for the volume of the cone

I have parted to make taking the derivative easier.

3) Take the derivative of with respect to h using the quotient rule.

4) Set and solve for h.

Note that the derivative does not exist at h=2x but 2x and 0 is not in the domain so the only possible value for h is 4x.

5) Solve for the volume of the cone representing all values in terms of .

Substituting in the numerical value of we have