Optimize a Cone of Volume 500cm^3?

Question

Requires minimum surface area

Answer 1

I'm presuming you want to minimise only the lateral

surface area, not the circular base as well.

Volume, V = πr^2h/3 = 500

Therefore, r^2 = 1500/(πh)

and, r = 10*sqrt[15/(πh)]

Lateral Surface Area, A = πr*sqrt(r^2 + h^2)

Substitute for r and r^2 :

A = π*10*sqrt[15/(πh)]*sqrt[1500/(πh) + h^2]

Simplifying to a form suitable for differentiating :

A = 10sqrt(15)h^(-1) * (1500 + πh^3)^(1/2)

Differentiate and set equal to zero, to get the minimum :

(I've skipped through to the simplified form)

dA/dh = 15πhsqrt(15)/sqrt(1500 + πh^3) -

10sqrt[15(1500 + πh^3)]/h^2 = 0

Solving for h gives :

h = 10*(3/π)^(1/3) ≈ 9.84745 cm

Substituting in the equation for r gives :

r = [10/sqrt(2)]*(3/π)^(1/3) ≈ 6.96320 cm

Note the interesting relations between

r, h and L (side length of cone) :

r = r*sqrt(1)

h = r*sqrt(2)

L = r*sqrt(3)