# Calculus

A rectangular beam is cut from a cylindrical log of radius 30 cm. The strength of a beam of width w and height h is proportional to wh^2. Find the width and height of the beam of maximum strength.

Is the width 45 and the height 20.

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asked by George
1. Hey, the height is surely more than the width for strength
w = 1/2 width
h = 1/2 height
15 = sqrt(w^2+h^2)
w^2+h^2 = 225
so
h^2 = 225 - w^2
then
strength = s = k w h^2 = k w(225-w^2)
find where ds/dw = 0
s = k (225 w - w^3)
ds/dw = k (225 - 3 w^2) =0
w^2 = 225/3 = 75
then h^2 = 225 -75 = 150
so
w = 5 sqt3 and width = 10 sqrt 3
h = 5 sqrt 6 and length = 10 sqrt 6
by the way
h/w = sqrt 2 = 1.41

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posted by Damon

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