BuyFind

Single Variable Calculus: Concepts...

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

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

Answer to Problem 42RE

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

  827(Volume ofsphere) .

Explanation of Solution

Given:

Radius of sphere is r.

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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