Sure, let’s solve this step-by-step to find the requested breadth.
The strength of a beam varies according to its characteristics such as breadth ([tex]\(b\)[/tex]), depth ([tex]\(d\)[/tex]), and length ([tex]\(l\)[/tex]). Given that the strength ([tex]\(S\)[/tex]) of a beam is directly proportional to the breadth, the square of its depth, and inversely proportional to its length, we can write:
[tex]\[ S \propto \frac{b \cdot d^2}{l} \][/tex]
### Given
- Beam 1:
- Length ([tex]\(l_1\)[/tex]) = 5 meters
- Breadth ([tex]\(b_1\)[/tex]) = 40 mm
- Depth ([tex]\(d_1\)[/tex]) = 100 mm
- Beam 2:
- Length ([tex]\(l_2\)[/tex]) = 3 meters
- Depth ([tex]\(d_2\)[/tex]) = 80 mm
- Breadth ([tex]\(b_2\)[/tex]) needs to be calculated
### Step-by-Step Calculation:
1. Strength Formula for Both Beams:
[tex]\[ S_1 = \frac{b_1 \cdot d_1^2}{l_1} \][/tex]
[tex]\[ S_2 = \frac{b_2 \cdot d_2^2}{l_2} \][/tex]
2. Equal Strength Condition:
Since the beams have equal strength, we set [tex]\(S_1 = S_2\)[/tex]:
[tex]\[ \frac{b_1 \cdot d_1^2}{l_1} = \frac{b_2 \cdot d_2^2}{l_2} \][/tex]
3. Substitute Given Values:
[tex]\[ \frac{40 \cdot (100)^2}{5} = \frac{b_2 \cdot (80)^2}{3} \][/tex]
4. Simplify Left-Hand Side:
[tex]\[ \frac{40 \cdot 10000}{5} = \frac{40 \cdot 2000} = 80000 \][/tex]
5. Set Up Equality:
[tex]\[ 80000 = \frac{b_2 \cdot 6400}{3} \][/tex]
6. Solve for [tex]\(b_2\)[/tex]:
[tex]\[ 80000 \times 3 = b_2 \cdot 6400 \][/tex]
[tex]\[ 240000 = b_2 \cdot 6400 \][/tex]
[tex]\[ b_2 = \frac{240000}{6400} \][/tex]
[tex]\[ b_2 = 37.5 \text{ mm} \][/tex]
Therefore, the breadth of the second beam must be [tex]\(37.5\)[/tex] mm to have an equal strength as the first beam.
