Question

Prove that a cone with least curved surface area and given volume has radius =

Answer 1

Let r and h be the radius and height of the cone respectively.

Volume V=13πr2h

=πk3(constant) r2h=k or h=kr2------(1)

Surface S=πrl=πr(h2+r2−−−−−−√)

h=kr2from (1)

S=πrk2r4+r2−−−−−−−√

=πrk2+r6−−−−−√r4

=πk2+r6−−−−−√r

Step 2:

dSdr=π[6r52r6+k2√×r−r6+k−−−−√.1r2]

=3r6−(r6+k2)r2(r6+k2−−−−−√)

=(2r6+k2)r2(r6+k2−−−−−√)

=k2=2r6

dSdr=0

Step 3:

dSdr changes sign from -ve to +ve as r increases through the point k2=2r6

⇒S is the least at this point.

From (1) k2=h2r4

h2r4=2r6

h2=2r2

h=2r2−−−√

h=r2–√