A right circular cone has a given curved surface A. Show that when its volume is maximum the ratio of the the height to the base radius is √2:1 ?

1 Answer
Mar 24, 2018

hr=21

Explanation:

.

The formula for the curved surface area of a cone which does not include the area of the base is:

A=πrl where r is the radius of the base and l is the lateral height (slant height) as shown in yellow below:

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From the right angle triangle shown, using Pythagoras' formule, we get:

l2=h2+r2

l=h2+r2

Therefore, the curved area A is:

A=πrh2+r2

The volume formula is:

V=13πr2h

Let's solve for h from the area formula and substitute it into the volume formula:

A2=π2r2(h2+r2)

A2=π2r2h2+π2r4

π2r2h2=A2π2r4

h2=A2π2r4π2r2

h=A2π2r4πr

V=13πr2A2π2r4πr

V=13rA2π2r4=13r(A2π2r4)12

To find the maximum volume, we take the derivative of the volume function and set it equal to 0:

dVdr=13[r(12)(A2π2r4)12(4π2r3)+(A2π2r4)12]

dVdr=13(2π2r4A2π2r4+A2π2r4)

dVdr=2π2r4+A2π2r43A2π2r4

dVdr=3π2r4+A23A2π2r4=0

3π2r4+A2=0

A2=3π2r4

Let's substitute for A from the area formula:

π2r2(h2+r2)=3π2r4

Dividing both sides by π2r2, we get:

h2+r2=3r2

h2=2r2

h2r2=2

hr=21