SOLUTION: The volume of a cylindrical can is 500 cubed cm. the material used to make the top and bottom costs .o12 cent/squared cm, the material used for the sides costs .01 cent/ squared cm

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Question 309449: The volume of a cylindrical can is 500 cubed cm. the material used to make the top and bottom costs .o12 cent/squared cm, the material used for the sides costs .01 cent/ squared cm, and the seam joining the top and bottom to the sides costs .015 cent/cm. what size can would cost the least to produce?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The can is a cylinder of radius R and length L.
V=pi%2AR%5E2%2AL=500
The total cost is the cost of the top and bottom plus the cost of the sidewall. The costs are based on total area.
The top and bottom are both circles of radius R.
+A=pi%2AR%5E2
The contribution to the total cost is,
+Ct%2BCb=2%2Api%2AR%5E2%2A0.012%0D%0ATHe+sidewall+is+a+rectangle+of+length+%7B%7B%7B2piR and width L.
+A=2%2Api%2ARL
Its contribution to the total cost is,
+Cs=2%2Api%2ARL%2A0.01
The final contributor to cost is the seam which is length L.
Its contribution is
+Cx=L%2A0.015
The total cost equation is then
+Ctot=Ct%2BCb%2BCs%2BCx=0.024%2Api%2AR%5E2%2B0.02%2Api%2ARL%2B0.015L+
Using the volume equation you can get L as a function of R.
pi%2AR%5E2%2AL=500
R%5E2=500%2F%28pi%2AL%29
Now substitute into the cost equation,
+Ctot=0.024%2Api%2AR%5E2%2B0.02%2Api%2ARL%2B0.015L+

+Ctot=12%2FL%2B0.7927%2Asqrt%28L%29%2B0.015L
Now you have total cost as the function of one variable.
To find the minimum cost, take the derivative and set it equal to zero.
dC%2Fdt=-12%2FL%5E2%2B%281%2F2%29%280.7927%29%2Fsqrt%28L%29%2B0.015=0
I solved this numerically in EXCEL and got highlight_green%28L=9.02%29
From that, then use the volume equation to find R.
R%5E2=500%2F%28pi%2AL%29
R%5E2=500%2F%28pi%2A9.02%29
highlight_green%28R=4.20%29