# The height of the pyramid of minimum volume whose base is a square and whose base triangular faces are all tangents to the 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 22P
To determine

## The height of the pyramid of minimum volume whose base is a square and whose base triangular faces are all tangents to the sphere.

Expert Solution

### Explanation of Solution

Given information:

The sphere has radius r. The base of the pyramid is a regular n-gon.

Calculations:

Let the height of the Pyramid for minimum volume be h

Radius of the sphere is r

Therefore,

h=r+x           Where, x= Distance between the center of sphere and cone of pyramid.

Let the side of the square base be 2b

Area of the square base is 4b2

Now from similar triangles rules we get,

bx+r=rx2r2b=r(x+r)x2r2

Now volume of the pyramid is,

v=13Ah=13(x+r).4.r2(x+r)2(x2r2)2v=4.r33(x+r)2(xr)

Now we differentiate to get the minimum value,

v'=4.r3.2.(x+r)(xr)(x+r)2.1(xr)2Now equating v'=0 ,we get,v'=4.r3.2.(x+r)(xr)(x+r)2.1(xr)2=0x=3rEarlier we assumed that,h=x+rNow putting the value of x we get,h=x+r=3r+rh=4r.

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