Question

The radius of a right circular cone is increasing at a rate of 1.5 in/s while its height is decreasing at a rate of 2.1 in/s. At what rate is the volume of the cone changing when the radius is 120 in. and the height is 107 in.

Answer 1

The volume of the cone is changing at a rate of approximately 8670.796 cubic inches per second.

Explanation:

Geometrically speaking, the volume of the right circular cone (
`V`), in cubic inches, is defined by the following formula:

`V = (1)/(3)\cdot \pi \cdot r^(2)\cdot h` (1)

Where:

`r` - Radius, in inches.

`h` - Height, in inches.

Then, we derive an expression for the rate of change of the volume (
`\dot V`), in cubic inches per second, by derivatives:

`\dot V = (1)/(3)\cdot \pi \cdot (2\cdot r\cdot h \cdot \dot r + r^(2)\cdot \dot h)` (2)

Where:

`\dot r` - Rate of change of the radius, in inches per second.

`\dot h` - Rate of change of the height, in inches per second.

If we know that
`r = 120\,in`,
`\dot r = 1.5\,(in)/(s)`,
`h = 107\,in` and
`\dot h = -2.1\,(in)/(s)`, then the rate of change of the volume is:

`\dot V \approx 8670.796\,(in^(3))/(s)`

The volume of the cone is changing at a rate of approximately 8670.796 cubic inches per second.