Question

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.

Answer 1

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:

$V = \pi\cdot r^{2}\cdot h$ (Eq. 1)

Where:

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

$h$ - 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:

$\Delta V = \frac{\partial V}{\partial r}\cdot \Delta r + \frac{\partial V}{\partial h}\cdot \Delta h$ (Eq. 2)

Where:

$\Delta r$ - Change in radius, measured in centimeters.

$\Delta h$ - Change in height, measured in centimeters.

$\frac{\partial V}{\partial r}$ - Partial derivative of volume in radius, measured in square centimeters.

$\frac{\partial V}{\partial h}$ - Partial derivative of volume in height, measured in square centimeters.

All partial derivatives are obtained, respectively:

$\frac{\partial V}{\partial r} = 2\pi\cdot r\cdot h$ (Eq. 3)

$\frac{\partial V}{\partial h} = \pi\cdot r^{2}$ (Eq. 4)

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

$\Delta V = 2\pi\cdot r\cdot h \cdot \Delta r+\pi\cdot r^{2}\cdot \Delta h$

$\Delta V =\pi\cdot r \cdot (2\cdot h\cdot \Delta r +r\cdot \Delta h)$ (Eq. 5)

If we know that $r = 40\,cm$, $h = 100\,cm$ and $\Delta r = \Delta h = 1\,cm$, the change in volume of the cylinder is approximately:

$\Delta V = \pi\cdot (40\,cm)\cdot [2\cdot (100\,cm)\cdot (1\,cm)+(40\,cm)\cdot (1\,cm)]$

$\Delta V = 9600\pi\,cm^{3}$

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