Asked by Creating a formula from 2 variables

Hi, I'm having trouble with this problem...

"If the strength of a rectangular beam of wood varies as its breadth and the square of its depth, find the dimensions of the strongest beam that can be cut out of a round log, diameter d. "

I can't come up with an equation with only 1 variable.

Help me, Please!!!
Many thanks, Stu.
Answered by levy
Im not really sure but i think its

360(d)(120)
Answered by Creating a formula from 2 variables
Hi Levy, any chance you could explain how you came up with that solution?
I believe I should find values for Breath and Depth with relation to diameter, but I am stuck for a way to do it.

Many thanks.
Answered by Damon
rectangle must fit in circle of diameter d
B =chord1 of circle = d sin (theta1/2)
H = chord 2 of circle = d sin( [pi-theta]/2)
because the angles subtended by the inscribed breadth and depth add up to a straight line. pi is straight line angle in radians.
S = k B H^2
S = k d sin (theta/2)*d^2 sin^2 ([pi-theta]/2)
let A = theta/2 to make it all easier
S = k d^3 sin A sin^2 (pi/2-A)
but sin (pi/20-A) = cos(A)
so
S = k d^3 sin(A) cos^2(A)
take derivative and set to 0
0 = dS/dA = k d^3[ -2 sin^2(A)cos(A)+ cos^3(A)]
2 sin^2 (A) = cos^2(A)
but cos^2(A) = 1 - sin^2(A)
2 sin^2 A = 1 - sin^2(A)
3 sin^2 (A) = 1
sin(A) = sqrt(3)/3 = about .577
so A is about 35.2 degrees
so
B = d sin (A) = .577 d
H = d sin(90-A) =d sin 54.8deg = .817 d
Answered by Creating a formula from 2 variables
Damon, you're a God-send!

If you're ever in North Wales I'll buy you a pint!

1x10^3 Thanks.

Stu.
There are no AI answers yet. The ability to request AI answers is coming soon!