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A cylinder shaped can needs to be constructed to hold 300 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 0.05 cents per square centimeter. Find the dimensions for the can that will minimize production cost.

Respuesta :

barnuts

The cost formula is as follows:

c = 2πrh * 0.02 + 2πr^2 * 0.05 
c = 0.04 πrh + 0.10 πr^2 

The formula for volume in terms of height is:
h = V/(πr^2) 
h = 300/(πr^2) 

Substituting this h into the cost formula:
c = 0.04 πr[300/(πr^2)] + 0.10 πr^2 
c = 12/r + 0.10 πr^2 
c' = - 12/r^2 + 0.20 πr 

Equating to zero to get minima:
- 12/r^2 + 0.20 πr  = 0 
r^3 = 12/(0.20 π) 
r = 2.673 cm 

h = 300/(π * 2.673^2) 
h = 13.365 cm 

So the radius must be 2.673 cm and height 13.365 cm

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