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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

Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.

To determine

To find:

The volume of the largest cone that is inscribed in a sphere of radius r.

Explanation

1) Formula:

i. Volume of the cone =13πx2h

ii. Power rule for xn is given by

ddxxn=nxn-1

3) Given:

Radius of the sphere is r, cone is inscribed in the sphere.

2) Calculation:

Consider the cone, inscribed in a circle of radius r, now draw a median from point A to side DC, ABC = 90° so that BC = BD.

Now P is the Centroid of triangle. Draw a line from P to C which is the radius of the circle so PC = r,

Side BC= x and PC is the radius of the circle so PC= r, PB=y, so that by Pythagoras theorem in PBCPC2=PB2+BC2

r2=y2+x2

Solve x2

x2=r2-y2

x=±r2-y2

And height of the cone h=r+y

Substitute these values in the volume of the cone.

Volume of the cone =13πx2h

V(y)=13π(r2-y2)(r+y)

V(y)=π3(r2-y2)(r+y)

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