Question

A.) The formula for the volume of a sphere is given below and the relationship between the radius r and the diameter d is r=d/2. Find the EXACT volume of the sphere in terms of the diameter d. Be sure to fully simplify the expression.

B.) What is the EXACT volume of the sphere when the diameter is 2/3 centimeter?

Answer 1

`\displaystyle V=(4)/(81)\cdot \pi\ cm^3`

Explanation:

The Volume of a Sphere

The volume of a sphere of radius r is given by:

`\displaystyle V=(4)/(3)\cdot \pi\cdot r^3`

A. )

If the diameter of the sphere is d, the radius is r=d/2. Substituting:

`\displaystyle V=(4)/(3)\cdot \pi\cdot \left ((d)/(2) \right )^3`

Operating

`\displaystyle V=(4)/(3)\cdot \pi\cdot (d^3)/(8)`

Simplifying

`\displaystyle V=(1)/(6)\cdot \pi\cdot d^3`

B.)

The diameter of the sphere is 2/3 cm, thus the volume is:

`\displaystyle V=(1)/(6)\cdot \pi\cdot \left ((2)/(3) \right )^3`

Operating:

`\displaystyle V=(1)/(6)\cdot \pi\cdot (8)/(27)`

`\displaystyle V=(8)/(27)(1)/(6)\cdot \pi`

`\boxed{\displaystyle V=(4)/(81)\cdot \pi\ cm^3}`