# Tofind: thevolume of the largest circular cone that can be inscribed in a sphere.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4, Problem 42RE
To determine

## Tofind:thevolume of the largest circular cone that can be inscribed in a sphere.

Expert Solution

TheVolume of the largest circular cone that can be inscribed in a sphere is

827(Volume ofsphere) .

### Explanation of Solution

Given:

Concept used:

Volume of circular cone:

V=13πr2h=π3(R2-x2)(R+x) .

If d2ydx2>0. the concave will be open upward and local minima can be found.

If d2ydx2<0. the concave will open downward and local maxima can be found.

Calculation:

Let r be the abase radius of x is the distance O the center of the sphere From the base and V the volume of the cone.

Height of the cone= R+x

V=13πr2h=π3(R2-x2)(R+x) .

π3(R2+R2x-Rx2-x2)

dVdx = π3(R2-2Rx-3x2) .

d2Vdx2=π3(-2R-6x) .

For max or min VdVdx=0 .

R2-2Rx-3x2 =0 .

(R+x)(x3x)=0 .

x=-R,x3 but x-R

When x=R3d2Vdx2<0 is max only when x=R3 .

Max V=13π(R2-R29)(R+R3)=32R381=827(43πR3)=827(Volume ofsphere) .

Hence the Volume of the largest circular cone that can be inscribed in a sphere is

827(Volume ofsphere) .

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