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

Answer:

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

Step-by-step explanation:

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

$V = \frac{1}{3}\cdot \pi \cdot r^{2}\cdot h$ (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:

$\dot V = \frac{1}{3}\cdot \pi \cdot (2\cdot r\cdot h \cdot \dot r + r^{2}\cdot \dot h)$ (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 $r = 120\,in$, $\dot r = 1.5\,\frac{in}{s}$, $h = 107\,in$ and $\dot h = -2.1\,\frac{in}{s}$, then the rate of change of the volume is:

$\dot V \approx 8670.796\,\frac{in^{3}}{s}$

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