Calculus homework quiz tommorow please help thanks

Question

A cylinder shaped can needs to be constructed to hold 450 cubic centimeters of soup. The material for the sides of the can costs 0.03 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.

Helpful information:

h : height of can, r : radius of can

Volume of a cylinder: `V = pi r^2 h`

Area of the sides: A = 2pirh

Area of the top/bottom: `A = pi r^2`

To minimize the cost of the can:

Radius of the can: __________________ 

Height of the can: __________________ 

Minimum cost: __________________ cents

Answer 1

Cost function

Cost(R,h) = 2*pi*R^2* 0.06 + 2*pi*R*H*0.03

= 0.12*pi*R^2 + 0.06*pi*R*h

However, the VOLUME is FIXED at 450...

pi*R^2*H = 450

H = 450/(pi*R^2)

substitutes...

0.12*pi*R^2 + 0.06*pi*R*(450)/(pi*R^2)

= 0.12*pi*R^2 + 0.06*450/R

= 0.12*pi*R^2 + 27/R

= Cost(R) is the cost function in terms of R

= C(r)

minimizing...

0= dC/dr = 0.24*pi*R - 27 *R^(-2)

0 = 0.24*pi*R^3 - 27

R^3 = 27/ (0.24*pi)

cube root...

R = 3/ (0.24*pi)^(1/3)

= 3 * (0.24*pi)^(2/3) / (0.24*pi)