STRENGTH OF BEAM ACROSS A RECTANGLE AREA OF A LOG

Question

A wooden beam has a rectangular cross section of height h in. and width w in. (see the accompanying figure). The strength S of the beam is directly proportional to its width and the square of its height. What are the dimensions of the cross section of the strongest beam that can be cut from a round log of diameter d = 29 in.? Hint: S = kh2w, where k is a constant of proportionality. (Round your answers to one decimal place.)
w =___in
h = ___in

Answer 1

S=kh²w and from the circular log w²+h²= d².

 

So, w = (d²-h²)^.5 ,substitute this into the equation for S ...

 

S = kh²(d²-h²)^.5  Now, differentiate this with respect to h...

 

dS/dh = [kh² (1/2 * (d²-h²)^-1/2 * (-2h)] + [(d²-h²)^1/2 * 2kh] 

 

          = -2kh³/(d²-h²)^.5  +  2kh(d²-h²)^.5 

 

set this equal to zero to find the critical points, and solve ..

 

2kh³/(d²-h²)^.5  =  2kh(d-h²)^.5

 

a little algebra gets you to ..

 

h³ = 2hd² -2h³   then to ..

 

3h³  - 2hd² = 0  factor out an h ..

 

h(3h² -2d²) = 0   now set (3h² -2d²) = 0 , a little more algebra we get...

 

h² = 2d²    so h = d(2/3)^.5    d = 29, so ..

 

h = 29 (2/3)^.5 = 23.68 in

 

w = (29² - 23.68²)^.5  = (280.257)^.5 = 16.74 in.