in question 2, we consider how to minimize the cost of manufacturing a (cylindri

Question

in question 2, we consider how to minimize the cost of manufacturing a (cylindrical) metal can with fixed volume V . Suppose the height of a can is given by h and the radius of its base is given by r. Note that real cans generally have a ratio of h/r varying from 2 to around 3.8.

2. (3 points)
Suppose we cut out the tops and bottoms of cans from squares of side 2r. Suppose also that we cut out rectangles from sheets of metal to make the sides of the can (which we assume leaves no waste). The process of cutting out tops and bottoms from squares of metal, however, leaves considerable waste metal. In this case, show that the amount of metal used for each can is minimized when h/r = 8/pi = 2.55

Explanation / Answer

V=R2H=C OR H=C/(R2), WHERE v is volume, r radius, h height. let s= surface area of side and 2 squares with sides of 2r to form circles

s=2rh +8r2 or eliminate h

s=2rc/(R2) +8r2

s=2cr-1 +8r2

s '= -2cr-2+16r =0 so r3=1/8 or r=(1/8)1/3

h/r=8/

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