A spnencal balloonI Is being inliated al a constant rate of 20 cubic inches per sect How last is the radlus of the Dalloon Cnanging at the instant the balloon s diameter is 12 inches? Is the radius changing more rapidly when d = 12 or when d = 16? Why? Draw several spheres with different radii, and observe that as volume changes, the radius, diameter, and surface area of the balloon also change. Recall that the volume of a sphere of radius r is V = Tr³. Note that in the setting of this problem, both V and r are changing as time t changes, and thus both V and r may be viewed as implicit functions of t, with respective derivatives and . Differentiate both sides of the equation V = Tr3 with respect to t (using the chain rule on the right) to find a formula for that depends on both r and AP dt 12.566r^2 At this point in the problem, by differentiating we have "related the rates" of change of V and r. Recall that we are given in the problem that the balloon is being inflated at a constant rate of 20 cubic inches per second. To which derivative does this rate correspond?

A spnencal balloonI Is being inliated al a constant rate of 20 cubic inches per sect How last is the radlus of the Dalloon Cnanging at the instant the balloon s diameter is 12 inches? Is the radius changing more rapidly when d = 12 or when d = 16? Why? Draw several spheres with different radii, and observe that as volume changes, the radius, diameter, and surface area of the balloon also change. Recall that the volume of a sphere of radius r is V = Tr³. Note that in the setting of this problem, both V and r are changing as time t changes, and thus both V and r may be viewed as implicit functions of t, with respective derivatives and . Differentiate both sides of the equation V = Tr3 with respect to t (using the chain rule on the right) to find a formula for that depends on both r and AP dt 12.566r^2 At this point in the problem, by differentiating we have "related the rates" of change of V and r. Recall that we are given in the problem that the balloon is being inflated at a constant rate of 20 cubic inches per second. To which derivative does this rate correspond?

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

ChapterP: Prologue: Calculator Arithmetic

Section: Chapter Questions

Problem 18E: When the Radius Increases a. A rope is wrapped tightly around a wheel with radius of 2 feet. If the...

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