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Old 12-09-2008   #1
Kelly101
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Unhappy Need help on calculus optimization problem!

I am having the hardest time figuring this out.


In developing a new juice product, a beverage company wishes to optimize the shape of the cylindrical can to minimize cost. The volume of the can must be 50 cubic centimeters. The side of the can are made out of a different material from the top and the bottom and so have different costs. The material for the sides costs $0.002 per square centimeter, and the material for the top and the bottom costs $0.003 per square centimeter. Find the dimensions of the can (radius and height) that minimize the cost of the can, and the minimum cost. Include a reasonable domain for your variable(s), and a proof that you have found the minimum. Round your final answer to two deciml places, and include units on all answers.
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Old 12-10-2008   #2
MAS1

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Quote:
Originally Posted by Kelly101 View Post
I am having the hardest time figuring this out.


In developing a new juice product, a beverage company wishes to optimize the shape of the cylindrical can to minimize cost. The volume of the can must be 50 cubic centimeters. The side of the can are made out of a different material from the top and the bottom and so have different costs. The material for the sides costs $0.002 per square centimeter, and the material for the top and the bottom costs $0.003 per square centimeter. Find the dimensions of the can (radius and height) that minimize the cost of the can, and the minimum cost. Include a reasonable domain for your variable(s), and a proof that you have found the minimum. Round your final answer to two deciml places, and include units on all answers.
Volume of a cylinder = PI*r^2*h = 50 cc
So, r^2 = 50/(PI*h) cm^2 and h = 50/(PI*r^2) cm

Area of side = 2*PI*r*h cm^2
Area of top and bottom (T&B) = 2*PI*r^2 cm^2

Cost of side = $(0.002)(Area of side) = $0.01257*r*h
Cost of T&B = $ (0.003)(Area of T&B) = $0.01885r^2

Total cost = 0.01257*r*h + 0.01885r^2

Then substituting for h (using the formula from the volume), I get

Total cost = .01257*r*(15.9155/r^2) + 0.01885*r^2

Total cost = 0.2/r + 0.01885*r^2

Take derivative with respect to r gives:
der(Cost) = -(0.2/r^2) + 0.0377*r

Set der(cost) = 0 gives 0.0377*r = 0.2/(r^2)

Then r^3 = 5.30504 so r = 1.74407 cm. Substituting back in gives h = 5.2323 cm.

I also tried it substituting for r instead of h and got the same values for r and h.

So the minimum cost = $0.17 (rounded to the nearest hundredth).

Does this sound right? Good Luck!
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