Max A. answered • 01/10/20
Professional Engineer with a Strong Tutoring/Academic Background
As a real life structural engineer, this problem sort of hits home for me so I will give it an answer. This is basically a proportional relationship problem. The most basic example is when two quantities vary directly with each other. y = kx, where "k" is a proportionality constant between the two variables. If I give a series of "x" values 1, 2, 3, 4, ..., and corresponding "y" values of 2, 4, 6, 8,...., we can see the two variables are directly proportional, and the constant of proportionality is k=2.
Let's expand on this definition for this problem. In this case, the maximum weight a beam can support (W) has a relationship with multiple variables. Specifically, it is directly proportional to the beam's width (b) and the square of the height (h2). It is also inversely proportional with the beam's length (L). We can write a single expression of the relationship below:
W = k*[(b*h2)/L]
From the first part of the problem, we know the following:
W = 20 tons (it is showing as "2020" in your question but I am assuming a realistic value)
b = 1/3 feet
h = 1/2 feet
L = 14 feet (same assumption as above)
The only thing we don't know is "k". So let's solve for it.
20 = k*[((1/3)*(1/2)2)/14]
20 = k*[((1/3)*(1/4))/14]
20 = k*[(1/12)/14]
20 = k*(1/168)
k = 3360
Now, let's look at the info in the second part of the problem. You'll see the only thing we are missing from our original equation is "W". This is exactly what we are solving for!
b = 1/2 feet
h = 1/3 feet
L = 12 feet
k = 3360 (from above)
W = 3360*[((1/2)*(1/3)2)/12]
W = 3360*[((1/2)*(1/9))/12]
W = 3360*[(1/18)/12]
W = 3360*(1/216)
W = 3360/216
W = 15.56 tons
