Question 164175: a cylindrical soda pop can of radius r and height h is to be manufactured to hold exactly 500 milliliters of liquid when completely full. A manufacturer wishes to find the dimensions of the can with the minimum surface area. Your task is to numerically, algebraically and graphically investigate this situation.
Determine the radius x and height h that yields a can of minimal surface area. Compute this minimal surface area.
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! a cylindrical soda pop can of radius r and height h is to be manufactured to hold exactly 500 milliliters of liquid when completely full. A manufacturer wishes to find the dimensions of the can with the minimum surface area. Your task is to numerically, algebraically and graphically investigate this situation.
Determine the radius x and height h that yields a can of minimal surface area. Compute this minimal surface area.
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Tough problem.
The volume is PI*r^2*h (r = radius, h= height) = 500 cc
The area is 2PI*r*h + 2PI*r^2 (the 1st term is the cylinder, the 2nd the top and bottom)
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Eliminate the h term in the area by solving for h in terms of r.
PI*r^2*h = 500
h = 500/(PI*r^2)
Sub for h in the eqn for area:
Area = 2PI*r + 2PI*r^2
Area = (1000/r) + 2PI*r^2
Differentiate with respect to r, and set = to 0.
(-1000/r^2) + 4PIr = 0
4PIr^3 = 1000
r^3 = 1000/4PI = 79.57747155
r = 4.30127 cm
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Does that make sense? Solve for h, and for the area.
I'm not sure what is meant by "solve numerically", maybe an Excel sheet?
I can graph it tomorrow, email me with any questions.
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