The strength of a beam is directly proportional to its width and the square of its depth but inversely proportional to its length. If a beam the that is 5 inches wide, 8 inches deep, 5 feet long can support weight of 384 pounds, how much could the same type of beam that is 6 inches wide, 6 inches deep, and 12 feet long support?

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.

Learn how to calculate beam strength using proportional relationships. Solve a problem where beam strength depends on width, depth squared, and inversely on length. Find out how to apply these concepts with specific calculations and unit conversions. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation