If the silver top is ten times more expensive as stainless steel, what are the most economical dimensions (height and radius) of the container if it is to...

Question

... Hold 314cm3.

The figure is a cylindrical figure and the sides are stainless steel except for the top.

Answer 1

Let:

 

C = cost of stainless steel

 

 

The container has a volume of 314 cm³.

 

πr²h = 314

 

 

Since we have a surface area of a cylinder,

 

C = Rπr² + H(2πrh)

 

where R and H is the cost of their respective area parts.

 

Substituting the volume into the C function.  This will allow us to get function C in terms of r.

 

C = Rπr² + H(2πr(314/πr²))

 

C = πr²R + 628r^-1 H

 

 

Finally, we find the derivative of C and set it equal to zero to find the value of r that will give us the minimum cost.

 

 

 

Answer 2

 V = 314 = Πr²h, so h = 314/Πr²

 

Cost, C = cost of surface area, but with one end x10

 

C = Πr² + 10Πr²  + 2Πrh   add like terms, and substitute for h:

 

C = 11Πr²  + 2Πr(314/Πr²)  =  11Πr²  + 628/r

 

Derivative C' = 22Πr  - 628/r² = 0 for minimum cost

 

Rearranging and simplifying, r³ = 628/22Π

 

So r = 2.087 cm, and h = 314/Π(2.087)² ;  h = 22.95 cm