# Calculus

Calculate the height of a cylinder of maximum volume that can be cut from a cone of height 20 cm and base radius 80 cm ,

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1. radius = r = 80 - (80/20)h
r = 80 - 4 h

v = pi r^2 h
v = pi (80-4h)^2 h
dv/dh = 0 for max or min
0 = (80-4h)^2 + h (2)(80-4h)(-4)
0 = 6400-640h+16h^2 - 640h+32h^2
48h^2 - 1280 h + 6400 = 0
6 h^2 - 160 h + 800 = 0
3 h^2 - 80 h+ 400 = 0
(3 h - 20 )(h - 20 ) = 0
h = 30 (no good, minimum)
or
h = 20/3 approximately 7

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