The strength of a rectangular beam is proportional to width*depth^2. What are the dimensions of the strongest rectangular beam that can be cut out of a 12 inch diameter log?

Answer 1

let the width be w, and the depth be d
then
S = kwd^2 where k is a constant
but d^2 + w^2 = 144
d^2 = 144-w^2

S = kw(144-w^2)
= 144kw - kw^3
d(S)/dw = 144k - 3kw^2
= 0 for max of S
3kw^2 = 144k
w^2 = 48
w = √48
then d = .....

I will let you finish up.