A cylinder shaped can needs to be constructed to hold 250 cubic centimeters of s

Question

A cylinder shaped can needs to be constructed to hold 250 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 .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 = pir^2h Area of the sides: A = 2pirh Area of the top/bottom: A = pir^2 To minimize the cot of the can: Radius of the can: heights of the can: Minimum cost:

Explanation / Answer

According to condition we will set up it like that ,

c = 2pi*r*h*0.04 + 2pi*r^2*0.06
c = 0.08pi*r*h + 0.12pi*r^2

h = v/(pi*r^2)
h = 250/(pi*r^2)

c = 0.08pi*r*250/(pi*r^2) + 0.12pi*r^2
c = 20/r + 0.12pi*r^2

c' = -20/r^2 + 0.24pi*r

-20/r^2 + 0.24pi*r = 0
r^3 = 20/(0.24pi)
r = 2.985 is the radius

h = 250/(pi*2.985^2)
h = 8.936 cm height

c = 20/2,985 + 0.12pi*2,985^2=6.70+3.357
c = 10.06 cents/cm^2   minimum cost

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