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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

A sphere of radius 1 overlaps a smaller sphere of radius r in such a way that their intersection is a circle of radius r. (In other words, they intersect in a great circle of the small sphere.) Find r so that the volume inside the small sphere and outside the large sphere is as large as possible.

To determine

To find:

Value of r

Explanation

1) Concept:

Use the equation for the volume of cap with height h and subtract the volume of cap of the larger sphere from the volume of hemisphere of the small sphere. Then maximize this function according to the first derivative test.

2) Calculation:

Let R=1 be the radius of the sphere.

Consider the portion shaded in gray color to be the cap of the sphere of height h.

Therefore, the volume of the cap of the sphere is

V=13πh2(3r-h)

To calculate h, consider the triangle ABC

1-h2+r2=1

1-h2=1-r2

1-h=1-r2

h=1-1-r2

Vcap=13π1-1-r223-1-1-r2

Vcap=13π1-21-r2+1-r22+1-r2

Vcap=13π2-21-r2-r22+1-r2

Vcap=13π4-41-r2-2r2+21-r2-21-r2-r21-r2

Vcap=13π4-41-r2-2r2+21-r2-2+2r2-r21-r2

Vcap=13π2-21-r2-r21-r2

According to the given condition, maximize the area of region shaded in red.

To find the volume of the shaded region, subtract volume of the cap of the large sphere from the volume of the hemisphere.

Volume of hemisphere=23πr3

V=Vhemisphere-Vcap

V=23πr3-π32-21-r2-r21-r2

V=π32r3-2-21-r2-r21-r2

V=π32r3-2+21-r2+r21-r2

V=π32r3-2+(2+r2)

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