The maximum weight that a rectangular beam can support varies jointly as its width and the square of its height and inversely as its length.

If a beam \(\frac{1}{3}\) foot wide, \(\frac{1}{2}\) foot high, and 13 feet long can support 10 tons, find how much a similar beam can support if the beam is \(\frac{2}{3}\) foot wide, \(\frac{1}{4}\) foot high, and 20 feet long.

The maximum weight is _____ tons. (Round to one decimal place as needed.)

The maximum weight a beam can support is calculated using the formula W = k imes rac{w imes h^2}{l}, where k is found using the first beam's dimensions and weight. This constant is then applied to the new beam's dimensions to find its maximum supported weight.

The maximum weight W that a rectangular beam can support varies jointly as its width w and the square of its height h, and inversely as its length l. This relationship can be represented by the formula W = k imes rac{w imes h^2}{l}, where k is the constant of proportionality.

First, to find the constant k using the given information about the first beam:

10 tons = k imes rac{1/3 ft imes (1/2 ft)^2}{13 ft}

Now we will solve for k and then use it to find the maximum weight the second beam can support, with the new dimensions provided:

Calculate the constant k using the given dimensions and maximum weight of the first beam.
Apply the constant k to the new beam dimensions to calculate the new maximum weight W.

Step-by-Step:

1. Find k:

10 = k imes rac{(1/3) imes (1/4)^2}{13}
So, k = 10 imes rac{13}{(1/3) imes (1/4)^2}

2. Calculate W for the new beam:

W = k imes rac{(2/3) imes (1/4)^2}{20}

After solving these equations, you'll find the maximum weight the second beam can support. Round the final answer to one decimal place as needed.