SOLUTION: A cylinder shaped can needs to be constructed to hold 500 cubic centimeters of soup. The material for the sides of the can costs 0.04 cents per square centimeter. The material for

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Question 1193900: A cylinder shaped can needs to be constructed to hold 500 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 0.07 cents per square centimeter. Find the dimensions for the can that will minimize production cost (round to 4 digits).
Helpful information:
h : height of can, r : radius of can
Volume of a cylinder: V=πr2h
Area of the sides: A=2πrh
Area of the top/bottom: A=πr2
To minimize the cost of the can:
Radius of the can:
Height of the can:
Minimum cost: cents
Answer by ikleyn(52914) About Me  (Show Source):
You can put this solution on YOUR website!
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A cylinder shaped can needs to be constructed to hold 500 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 0.07 cents per square centimeter.
Find the dimensions for the can that will minimize production cost (round to 4 digits).
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r and h are in centimeters.

Cost for the top and bottom materials, total is  0.07%2A%28pi%2Ar%5E2%29 + 0.07%2A%28pi%2Ar%5E2%29 = 0.14%2Api%2Ar%5E2 cents.

Cost for the lateral surface is  0.04%2Api%2Ar%2Ah cents.


The function to minimize is  F(r,h) = 0.14%2Api%2Ar%5E2+%2B+0.04%2Api%2Ar%2Ah.    (1)


The restriction is  pi%2Ar%5E2%2Ah = 500      (2)   (the volume).


From the restriction,  h = 500%2F%28pi%2Ar%5E2%29.    (3).


Substitute (3) into (2).  Then the function to minimize is

    f(r) = 0.14%2Api%2Ar%5E2+%2B+0.04%2Api%2Ar%2A%28500%2F%28pi%2Ar%5E2%29%29 = 0.14%2Api%2Ar%5E2+%2B+20%2Fr.


To find the minimum of f(r), take the derivative and equate it to zero.  You will get

    f'(r) = 0.28%2Api%2Ar - 20%2Fr%5E2 = 0,  

which implies  

    0.28%2Api%2Ar%5E3 = 20,  r%5E3 = 20%2F%280.28%2Api%29 = 20%2F%280.28%2A3.14%29 = 22.74795,

    r = root%283%2C22.74795%29 = 2.83 cm.


Then h = 500%2F%28pi%2Ar%5E2%29 = 500%2F%283.14%2A2.83%5E2%29 = 19.88 cm.


ANSWER.  r = 2.83 cm;  h = 19.88 cm.

Solved.