Asked by Chidera
A rectangular beam is to be cut from the log into a circular cross section. If the strength of the beam is proportional to the width and the square of the depth, find the dimensions that will give the strongest beam
Answers
Answered by
oobleck
I assume we can center the beam in the circle. So, let
2x = width
2y = depth
That means that x^2+y^2 = r^2, the radius of the log
If we let the strength be z, then we know that
z = kxy^2 = kx(r^2-x^2) = kr^2x - kx^3
dz/dx = k(r^2 - 3x^2)
Now just find x where dz/dx=0
You can see that z will be a maximum.
2x = width
2y = depth
That means that x^2+y^2 = r^2, the radius of the log
If we let the strength be z, then we know that
z = kxy^2 = kx(r^2-x^2) = kr^2x - kx^3
dz/dx = k(r^2 - 3x^2)
Now just find x where dz/dx=0
You can see that z will be a maximum.
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