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Answer :
To solve this problem, we need to understand how the weight that a beam can support varies with its dimensions. The maximum weight that a rectangular beam can support is determined by the following relationships:
- It varies jointly as the width of the beam.
- It varies as the square of the beam's height.
- It varies inversely with the length of the beam.
Let's denote:
- [tex]\( W \)[/tex] as the weight the beam can support,
- [tex]\( w \)[/tex] as the width of the beam,
- [tex]\( h \)[/tex] as the height of the beam,
- [tex]\( l \)[/tex] as the length of the beam.
The mathematical expression of these relationships is:
[tex]\[ W = k \cdot w \cdot h^2 \cdot \frac{1}{l} \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
Step 1: Determine the constant of proportionality, [tex]\( k \)[/tex], using the first beam's data.
- Width, [tex]\( w_1 = \frac{1}{3} \)[/tex] foot
- Height, [tex]\( h_1 = \frac{1}{4} \)[/tex] foot
- Length, [tex]\( l_1 = 15 \)[/tex] feet
- Weight supported, [tex]\( W_1 = 14 \)[/tex] tons
Plug these values into the equation:
[tex]\[ 14 = k \cdot \frac{1}{3} \cdot \left(\frac{1}{4}\right)^2 \cdot \frac{1}{15} \][/tex]
Simplify:
[tex]\[ 14 = k \cdot \frac{1}{3} \cdot \frac{1}{16} \cdot \frac{1}{15} \][/tex]
[tex]\[ 14 = k \cdot \frac{1}{720} \][/tex]
To find the constant [tex]\( k \)[/tex]:
[tex]\[ k = 14 \times 720 = 10080 \][/tex]
Step 2: Calculate the weight that the second beam can support using the same formula.
For the second beam:
- Width, [tex]\( w_2 = \frac{1}{4} \)[/tex] foot
- Height, [tex]\( h_2 = \frac{1}{2} \)[/tex] foot
- Length, [tex]\( l_2 = 20 \)[/tex] feet
We now know that [tex]\( k = 10080 \)[/tex], so:
[tex]\[ W_2 = 10080 \cdot \frac{1}{4} \cdot \left(\frac{1}{2}\right)^2 \cdot \frac{1}{20} \][/tex]
Calculate the weight:
[tex]\[ W_2 = 10080 \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{20} \][/tex]
[tex]\[ W_2 = 10080 \cdot \frac{1}{320} \][/tex]
[tex]\[ W_2 = 31.5 \][/tex]
Therefore, the second beam can support 31.5 tons.
- It varies jointly as the width of the beam.
- It varies as the square of the beam's height.
- It varies inversely with the length of the beam.
Let's denote:
- [tex]\( W \)[/tex] as the weight the beam can support,
- [tex]\( w \)[/tex] as the width of the beam,
- [tex]\( h \)[/tex] as the height of the beam,
- [tex]\( l \)[/tex] as the length of the beam.
The mathematical expression of these relationships is:
[tex]\[ W = k \cdot w \cdot h^2 \cdot \frac{1}{l} \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
Step 1: Determine the constant of proportionality, [tex]\( k \)[/tex], using the first beam's data.
- Width, [tex]\( w_1 = \frac{1}{3} \)[/tex] foot
- Height, [tex]\( h_1 = \frac{1}{4} \)[/tex] foot
- Length, [tex]\( l_1 = 15 \)[/tex] feet
- Weight supported, [tex]\( W_1 = 14 \)[/tex] tons
Plug these values into the equation:
[tex]\[ 14 = k \cdot \frac{1}{3} \cdot \left(\frac{1}{4}\right)^2 \cdot \frac{1}{15} \][/tex]
Simplify:
[tex]\[ 14 = k \cdot \frac{1}{3} \cdot \frac{1}{16} \cdot \frac{1}{15} \][/tex]
[tex]\[ 14 = k \cdot \frac{1}{720} \][/tex]
To find the constant [tex]\( k \)[/tex]:
[tex]\[ k = 14 \times 720 = 10080 \][/tex]
Step 2: Calculate the weight that the second beam can support using the same formula.
For the second beam:
- Width, [tex]\( w_2 = \frac{1}{4} \)[/tex] foot
- Height, [tex]\( h_2 = \frac{1}{2} \)[/tex] foot
- Length, [tex]\( l_2 = 20 \)[/tex] feet
We now know that [tex]\( k = 10080 \)[/tex], so:
[tex]\[ W_2 = 10080 \cdot \frac{1}{4} \cdot \left(\frac{1}{2}\right)^2 \cdot \frac{1}{20} \][/tex]
Calculate the weight:
[tex]\[ W_2 = 10080 \cdot \frac{1}{4} \cdot \frac{1}{4} \cdot \frac{1}{20} \][/tex]
[tex]\[ W_2 = 10080 \cdot \frac{1}{320} \][/tex]
[tex]\[ W_2 = 31.5 \][/tex]
Therefore, the second beam can support 31.5 tons.
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