CALCULUS HW PLEASE HELP!!! ASAP NEED HELP ON HW THANKS SOOO MUCH!!

Question

A cylinder shaped can needs to be constructed to hold 600 cubic centimeters of soup. The material for the sides of the can costs 0.02 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.06 cents per square centimeter. Find the dimensions for the can that will minimize production cost.

FIND height, cost, and radius

Answer 1

Volume of a cylinder V = πr²h where r≡radius and h≡height

h = V/(πr²) = 600/(πr²)

Surface area = 2πr² + 2πrh (Twice top and bottom and the side surface areas)

Cost, C = 0.06(2)πr² + 0.02(2πrh)

Substituting for h

C(r) = 0.12πr² + 0.04πr(600/πr²) = 0.12πr² + 24/r

To minimize the cost, take the derivative of the cost, set that to zero and solve for r.

C'(r) = 0.24πr - 24/r²

0.24πr = 24/r²

r³ = (24/0.24)/π = 100/π

r = (100/π)^1/3 = 3.16 cm

h = 600/(π(3.16)²) = 19 cm

C = 0.12πr² + 24/r = 3.76 + 7.59 = 11.35 cents