Let V be the volume of a cylinder having height h and radius...

Let $V$ be the volume of a cylinder having height $h$ and radius $r,$ and assume that $h$ and $r$ vary with time. (a) How are $d V / d t, d h / d t,$ and $d r / d t$ related? (b) At a certain instant, the height is 6 in and increasing at 1 in $/ \mathrm{s}$, while the radius is 10 in and decreasing at $1 \mathrm{in} / \mathrm{s}$. While the radius is $10 \mathrm{in}$ and decreasing at $1 \mathrm{in} / \mathrm{s}$. How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant?

Let $V$ be the volume of a cylinder having height $h$ and
radius $r,$ and assume that $h$ and $r$ vary with time.
(a) How are $d V / d t, d h / d t,$ and $d r / d t$ related?
(b) At a certain instant, the height is 6 in and increasing
at 1 in $/ \mathrm{s}$, while the radius is 10 in and decreasing
at $1 \mathrm{in} / \mathrm{s}$. While the radius is $10 \mathrm{in}$ and decreasing at $1 \mathrm{in} / \mathrm{s}$. How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant?

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Key Concepts

Cylinder Volume Formula

This concept involves the geometric formula for the volume of a cylinder, given by V = ? r² h. Understanding this formula is essential when modeling the quantity of a substance or material as a function of time in problems involving cylindrical objects.

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time by relating it to other quantities whose rates of change are known. In this context, the volume of the cylinder changes over time because both its height and radius vary with time.

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. When the variables in the volume formula (r and h) are functions of time, the chain rule allows us to compute dV/dt by taking the derivative of V with respect to these variables and then multiplying by their respective time derivatives.

Product Rule

The product rule is a rule of differentiation used when taking the derivative of a product of two or more functions. In finding the derivative of the cylinder's volume, both the square of the radius and the height are functions of time, so the product rule is applied while differentiating the product ? r² h with respect to time.

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