Let

y = the strength of a beam

x = the length

The strength of a beam varies inversely with the square of its length means:

y = k / x^2

where k is a proportionality constant.

A 10 foot beam can support 400 pounds means:

400 = k / 10^2

k/100 = 400 multiply both sides by 100

k = 400*100

k = 40,000

The equation for the strength of the beam is:

y = 40,000/x^2

We want to find y when x = 12 ft:

y = 40,000/12^2

y = 278 pounds

A 12 foot beam can support 278 pounds.