Table of contents

4. Applications of Derivatives

Implicit Differentiation

Problem 3.8.44b

Textbook Question

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.
b. Evaluate this derivative when a=6 and b=10.

Verified step by step guidance

Identify the formula for the volume of the torus: V = \frac{\pi^2 (b + a)(b - a)^2}{4b}.

Substitute the given values a = 6 and b = 10 into the volume formula to express V in terms of these constants.

Differentiate the volume function V with respect to a and b to find the derivative dV/da and dV/db.

Evaluate the derivatives at the specific values a = 6 and b = 10 to find the rate of change of volume with respect to the inner and outer radii.

Interpret the results of the derivatives to understand how changes in the radii affect the volume of the torus.

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Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.b. Evaluate this derivative when a=6 and b=10.