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:

Bulimia Nervosa

An eating disorder marked by binge eating followed by methods to avoid weight gain such as vomiting, fasting, or excessive exercise.

Mood Disorders

Psychological disorders characterized by the elevation or lowering of a person's mood, such as bipolar disorder or major depression.

Opioid Abuse

The misuse of opioid drugs, including prescription painkillers and illicit substances like heroin, often leading to addiction, health problems, or overdose.

Bulimia Nervosa

An eating disorder characterized by episodes of binge eating followed by behaviors such as vomiting to prevent weight gain.