The strength, S, of a rectangular wooden beam is proportional to its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a 12 inch diameter cylindrical log.

Answer 1

let the width of the beam be 2w and the depth be 2d

S = k(2w)(4d^2)
= 8k w(d^2)

I drew a circle and incribed a rectangle with the defined dimensions.
Notice the diagonal is 12
Construct a righ-angled triangle, with sides w, d, and hypotenuse 6
w^2 + d^2 = 36
d^2 = 36-w^2

in S = 8k w(d^2)
= 8k w(36-w^2)
= 8k (36w - w^3)

dS/dw = 8k(36-3w^2) , remember k is a constant
= 0 for a max of S
3w^2 - 36 = 0
w^2 = 12
w = √12 = 2√3
d^2 = 36-12 = 24
d = 2√6

so the width is 4√3 and the depth is 4√6

Answer 2

Do you mean 2√3 x 2√6?

Answer 3

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