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
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\times \quad 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 \(W = k * (w * h^2) / L\), 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 \(k = W1 * L1 / (w1 * h1^2) = 84.0\).

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 \(W2 = k * (w2 * h2^2) / L2 = 18.0\) 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 \(\boxed{18}\) tons.