a cylinder shaped can needs to be constructed to hold 400 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 needs to be thicker, and costs 0.08 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

A cylinder can be seen as a collection of multiple congruent disks, stacked one above the other. In order to calculate the space occupied by a cylinder, we calculate the space occupied by each disk and then add them up.

Thus, the volume of the cylinder can be given by the product of the area of base and height.

Volume of cylinder = πr²h

According to the given question:

We have V cylinder = V = 400cc

We will first find the areas involved

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 cylinder as function of x

V = πx²h ⇒ h= v/πx² ⇒ h = 400/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 + 800/x

Taking derivatives:

A´(x) = 4πx - 800/x²

A´(x) = 0 4πx - 800/x² =0 ⇒ πx - 200/x² = 0

( πx³ - 200 )/ x² = 0

πx³ - 200 = 0

x = 4 cm

and h = 400/πx² ⇒ h = 78.6 cm

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