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A circular-cylindrical oil drum is required to have a given surface area S
(including its lid and base), ie S is a constant.
Find the proportions of the design which contain the greatest volume V .
Surface area: S = 2(pi*r^2) + (2*pi*h)
Volume: V = pi*r^2*h
Find the relationship between the surface area and the volume
S = 2*(h+r)
Assume a value: S = 120
2h + 2r = 120
h + r = 60
h = (60-r)
V = pi*r^2*(60-r): replace h with (60-r)
Find Max volume. graph the equation y = 3.14*x^2*(60-x)
Max volume occurs when r = 40
then h = 60 - 40
h = 20
we can say max volume when radius:height = 2:1