A cylinder shaped can needs to be constructed to hold 200 cubic centimeters of soup. The material for the sides of the can costs 0.04 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.08 cents per square centimeter. Find the dimensions for the can that will minimize production cost.

x base radius = 3.169 cm

h height of cylinder = 6.34

Step-by-step explanation:

We have

V cylinder = V = 200 cc

We wll fnd frst the areas involve

Let x = radius of base then area of the base = π*x² and this is the area of the top too

For lateral area we need to get h the height of the clinder as fuction of x

V = πx²h ⇒ h = v/πx² ⇒ h = 200/πx²

Now the total area of the cylinder is:

A (x) = Area of the base + area of the top + lateral area

A (x) = 2*π*x² + 2πx h ⇒ A (x) = 2*π*x² + 2πx (V/πx²)

A (x) = 2*π*x² + 400/x

Taking derivatives:

A' (x) = 4πx - 400/x²

A' (x) = 0 4πx - 400/x² = 0 ⇒ πx - 100/x² = 0

(πx³ - 100) / x² = 0 πx³ - 100 = 0 x³ = 100/3.14 x = 3.169 cm

and h = 200/πx² ⇒ h = 6.34 cm