A paper cup is to be designed in the shape of a right circular cone. It must have a capacity of 12 fluid ounces (1 fluid ounce = 1.80469 cubic inches) of soft drink but it must use a minimum amount of material in its construction. What should the dimensions of this paper cup be and how much material is needed for its construction?

Question

a) One will need to compute derivatives and find and test critical values to obtain the answer.

Answer 1

a) is correct. Did you do that?

v = 1/3 pi r^2 h
12*1.80469 = pi/3 r^2 h
h = 64.9688/(pi*r^2)

surface area of cone is pi r √(r^2+h^2)
a = pi *r √(r^2+(64.9688/(pi*r^2))^2)
= pi √(r^6 + 427.671)/r

Now find da/dr

da/dr = (6.283r^6 - 1343.57)/[r^2 √(r^6 + 427.671)]

The denominator is never zero, so we just need to have
6.283r^6 = 1343.57
r = 2.445 in
h = 3.459 in

Better check my math. That's a strange-shaped paper cup!