In packaging a product in a can the shape of
right circular cylinder, various factors such as tradition and
supposed customer preferences may enter into decisions about what
shape (e.g. short and fat vs. tall and skinny) can might be used
for a fixed volume. Note, for example, all 12 oz. soda cans have
the same shape -- a height of about 5 inches and a radius of about
1.25 inches. Why?
What if the producers decison was based on minimizing the material
used to make the can? This would mean that for a fixed volume
V the shape of the can (e.g. the radius and the height)
would be determined by the minimum surface area for the can. What
is the relationship between the radius and the height in order
to minimize the surface area for a fixed volume?
What would be the shape of a 12 ounce soda can that minimizes
the amount of aluminum in the can?
Note: You undoubtedly have seen this problem before
in calculus. Do it WITHOUT calculus.
Mininum Surface - Picture
and formulas
Try using a spreadsheet. Fix V at some constant.
Make a column for r and compute a column for h. Make a graph of
S as a function of r or h.
Want to see a Graph of 12 oz can surface
area as a function of the radius?
Try using the arithmetic mean-- geometric mean
inequality.
Want to see a solution via the Arithmetic
Mean -- Geometric Mean Inequality?
Give up? Here's the
Bottom Line
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