Asked by George
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.
Is the width 45 and the height 20.
Answers
Answered by
Damon
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
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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