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 $1 \mathrm{in} / \mathrm{s}$, while the radius is 10 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 $1 \mathrm{in} / \mathrm{s}$, while the radius is 10 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

Implicit Differentiation

Implicit differentiation is a technique used when a relationship between variables is given in an implicit form rather than solved explicitly for one variable. In related rates contexts, it enables the differentiation of both sides of an equation with respect to time, connecting the rates of change of several interdependent variables.

Volume of a Cylinder

The volume of a cylinder is given by the formula V = ?r²h, where r is the radius and h is the height. This formula is central to problems involving cylinders with time-varying dimensions, as it provides the relationship between the geometric quantities that change over time.

Related Rates

Related rates problems involve finding the rate at which one quantity changes based on the rate at which another related quantity changes. These problems typically involve differentiating an equation that relates the various quantities with respect to time, thereby connecting their rates of change.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. In related rates problems, it is applied to differentiate functions where variables depend on time, allowing us to express the derivative of a function with respect to time in terms of the derivatives of its individual variables.

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