A right circular cylinder has a radius of cm and a height of cm. Use differentials to estimate the change in volume of the cylinder if its height and radius are both increased by cm. Give an exact answer.

The change in volume of the cylinder is 9600π cubic centimeters.

Step-by-step explanation:

Statement is incomplete. Complete statement is:

A right circular cylinder has a radius of 40 cm and a height of 100 cm. Use differentials to estimate the change in volume of the cylinder if its height and radius are both increased by 1 cm.

From Geometry we know that volume of right circular cylinder, measured in cubic centimeters, is represented by the following expression:

(Eq. 1)

Where:

- Radius of the right circular cylinder, measured in centimeters.

- Height of the right circular cylinder, measured in centimeters.

The change in volume of the cylinder, measured in cubic centimeters, is obtained by total differentials:

(Eq. 2)

Where:

- Change in radius, measured in centimeters.

- Change in height, measured in centimeters.

- Partial derivative of volume in radius, measured in square centimeters.

- Partial derivative of volume in height, measured in square centimeters.

All partial derivatives are obtained, respectively:

(Eq. 3)

(Eq. 4)

By applying (Eqs. 3, 4) in (Eq. 2), we obtain the resulting expression:

(Eq. 5)

If we know that , and , the change in volume of the cylinder is approximately:

The change in volume of the cylinder is 9600π cubic centimeters.