Surface-area growth of an expanding sphere

Question

The radius of a spherical bubble grows at a constant $\dfrac{9}{4\pi}$ inches per second. At the instant its volume is growing at $36$ cubic inches per second, how fast is its surface area growing, in square inches per second?

Short answer

$36$ . $V'=4\pi r^2r'=9r^2=36$ gives $r=2$ . Then $S'=8\pi rr'=8\pi\cdot2\cdot\tfrac{9}{4\pi}=36$ .

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Lessons behind this question

The Derivative as a Limit: Rates of Change and the Derivative Function
Transforms and Moment Generating Functions
The Chain Rule

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