The Volume of a Right Circular Cone of Radius $r$ And Height

question 25

Essay

The volume of a right circular cone of radius $r$ and height $h$ is $V = \frac { \pi } { 3 } r ^ { 2 } h$ . Suppose that the radius and height of the cone are changing with respect to time $t$ .
a. Find a relationship between $\frac { d V } { d t } , \frac { d r } { d t }$ , and $\frac { d h } { d t }$ .
b. At a certain instant of time, the radius and height of the cone are 12 in. and 13 in. and are increasing at the rate of $0.2 \mathrm { in } . / \mathrm { sec }$ and $0.5 \mathrm { in } . / \mathrm { sec }$ , respectively. How fast is the volume of the cone increasing?

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Definitions:

Excise Taxes

Taxes imposed on specific goods or services, such as tobacco or alcohol, typically aimed at reducing consumption of these goods or raising revenue.

Inelastic Demand

Inelastic demand refers to a market condition where the quantity demanded of a good or service changes very little in response to price changes.

Excise Tax

A tax on the production, sale, or consumption of a particular good or service within a country.

Efficiency

The optimal use of resources to achieve the best possible outcome or output with the least waste of time and effort.