A cylindrical log is to b cut so that it will yield a beam t

A cylindrical log is to b+ cut so that it will yield a beam that has a rectangular cross the width and the cube of the depth the diameter x of the long is 22 inches. what will depth the beam?

Solution

If we abbreviate \"stiffness\" as \"s\", and
use an arbitrary constant \"K\" as the
proportionality constant, then we have:
s = K*w*d^3

w^2 +d^2 = x^2 ---> w = sqrt(484 - d^2)

s =K*d^3*sqrt(484 - d^2)

stiffness: maximum stiffness , ds/d(d) = ( -d^2 + 484)^3/2 -3d^2(sqrt( -d^2 +484)

ds/d(d) =0

maximum occurs at d =11sqrt3 ;d = 19.05

depth = 19.05 inch; width = sqrt(484 -d^2) = 11 inch