Question

show that the right circular cone of least curved surface area and given volume has an altitude equal to root 2 times the radius of the base.

Answer 1

Question:

show that the right circular cone of least curved surface area and given volume has an altitude equal to root 2 times the radius of the base?

Answer:

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

Volume V=1/3 πr2h

=πk/3 -- (constant) r2h=k

or

h=kr2 --- (i)

Surface S = πrl = πr ( √h + √r2)

h=kr/ 2 ---- from (1)

S = πr √k2 / r4 + r2

=πr √k2+r6 / r4

=π √k2 + r6 / r

DS / DR 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 = r √2

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