Ritional Expressions
Word probtem on combined variation

Suppose that the maximum weight that a certain type of rectangular beam can support varies inversely as its length and iointiy as its width and the square of its height. Suppose also that a beam 6 inches wide, 2 inches high, and 12 feet long can support a maximum of 14 tons. What is the maximum weight that could be supported by a beam that is 4 inches wide, 3 inches high, and 14 feet long?
tons
$$\times 6$$

Step 1 ：Given that the maximum weight (W) that a beam can support varies inversely with its length (L) and jointly with its width (w) and the square of its height (h^2). This can be represented by the equation $W=k\ast (w\ast h^2)/L$, where k is the constant of variation.

Step 2 ：We are given that a beam 6 inches wide, 2 inches high, and 12 feet long can support a maximum of 14 tons. We can use these values to find the value of k. Given $w1=6$, $h1=2$, $L1=144$ (12 feet = 144 inches), and $W1=14$, we can substitute these into the equation to get $k=W1\ast L1/(w1\ast h{1}^2)=84.0$.

Step 3 ：We are asked to find the maximum weight that could be supported by a beam that is 4 inches wide, 3 inches high, and 14 feet long. Given $w2=4$, $h2=3$, and $L2=168$ (14 feet = 168 inches), we can substitute these values and the value of k into the equation to get $W2=k\ast (w2\ast h{2}^2)/L2=18.0$ tons.

Step 4 ：Final Answer: The maximum weight that could be supported by a beam that is 4 inches wide, 3 inches high, and 14 feet long is $\boxed{{\displaystyle 18}}$ tons.