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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.

User Marimba
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Answer:

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.

User Jonas Sourlier
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