To solve this problem, we'll use the relationship given in the problem, which states that the strength $S$ of a beam is directly proportional to its width $w$ and the square of its depth $d^2$, but inversely proportional to its length $L$.

Mathematically, this can be expressed as: $S \propto \frac{w \cdot d^2}{L}$ Or, introducing a constant of proportionality $k$: $S = k \cdot \frac{w \cdot d^2}{L}$

Step 1: Find the constant $k$

We know the strength $S$ for the first beam:

• $w = 5$ inches
• $d = 8$ inches
• $L = 5$ feet (which is $60$ inches, because $1$ foot = $12$ inches)
• $S = 384$ pounds

Substitute these values into the formula: $384 = k \cdot \frac{5 \cdot 8^2}{60}$

Now, solve for $k$: $384 = k \cdot \frac{5 \cdot 64}{60}$ $384 = k \cdot \frac{320}{60}$ $384 = k \cdot \frac{16}{3}$ $k = 384 \cdot \frac{3}{16}$ $k = 72$

Step 2: Calculate the strength of the second beam

Now, for the second beam:

• $w = 6$ inches
• $d = 6$ inches
• $L = 12$ feet (which is $144$ inches)

Substitute these values and the constant $k$ into the formula: $S = 72 \cdot \frac{6 \cdot 6^2}{144}$ $S = 72 \cdot \frac{6 \cdot 36}{144}$ $S = 72 \cdot \frac{216}{144}$ $S = 72 \cdot 1.5$ $S = 108 \, \text{pounds}$

Final Answer:

The second beam can support 108 pounds.

Do you want more details or have any questions?

Here are some related questions:

1. How would the result change if the length of the second beam was halved?
2. What happens to the strength if the depth of the second beam is increased by 2 inches?
3. How does the strength change if both width and depth are doubled?
4. What if the proportionality constant $k$ were different?
5. How does the strength formula relate to real-life materials like wood or steel?

Tip: When dealing with proportions, it's important to carefully convert all units to be consistent (e.g., inches or feet) before performing calculations.