Answer :
To solve this problem, we need to understand how the maximum weight a rectangular beam can support varies. The weight varies jointly as the width and the square of its height, and inversely as its length.
Here's how you can approach it:
1. Identify the formula:
The formula that describes the relationship is:
[tex]\[
W = k \cdot \frac{\text{width} \cdot (\text{height})^2}{\text{length}}
\][/tex]
where [tex]\( W \)[/tex] is the maximum weight the beam can support, and [tex]\( k \)[/tex] is a constant of proportionality.
2. Determine the constant [tex]\( k \)[/tex] using the first beam:
For the first beam, we have:
- Width = [tex]\( \frac{1}{3} \)[/tex] foot
- Height = [tex]\( \frac{1}{3} \)[/tex] foot
- Length = 10 feet
- Weight = 20 tons
Plug these values into the formula to solve for [tex]\( k \)[/tex]:
[tex]\[
20 = k \cdot \frac{\frac{1}{3} \cdot \left(\frac{1}{3}\right)^2}{10}
\][/tex]
Simplifying inside the fraction:
[tex]\[
20 = k \cdot \frac{\frac{1}{3} \cdot \frac{1}{9}}{10}
\][/tex]
[tex]\[
20 = k \cdot \frac{1}{27 \cdot 10}
\][/tex]
[tex]\[
20 = k \cdot \frac{1}{270}
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = 20 \times 270 = 5400
\][/tex]
3. Calculate the maximum weight for the second beam:
Now, use the same formula to find the weight the second beam can support with the given dimensions:
- Width = [tex]\( \frac{2}{3} \)[/tex] foot
- Height = [tex]\( \frac{1}{2} \)[/tex] foot
- Length = 18 feet
The formula becomes:
[tex]\[
W = 5400 \cdot \frac{\frac{2}{3} \cdot \left(\frac{1}{2}\right)^2}{18}
\][/tex]
Calculate the fraction:
[tex]\[
W = 5400 \cdot \frac{\frac{2}{3} \cdot \frac{1}{4}}{18}
\][/tex]
[tex]\[
W = 5400 \cdot \frac{\frac{2}{12}}{18}
\][/tex]
[tex]\[
W = 5400 \cdot \frac{1}{108}
\][/tex]
Simplifying:
[tex]\[
W = 50
\][/tex]
Thus, the maximum weight the second beam can support is 50.0 tons.
Here's how you can approach it:
1. Identify the formula:
The formula that describes the relationship is:
[tex]\[
W = k \cdot \frac{\text{width} \cdot (\text{height})^2}{\text{length}}
\][/tex]
where [tex]\( W \)[/tex] is the maximum weight the beam can support, and [tex]\( k \)[/tex] is a constant of proportionality.
2. Determine the constant [tex]\( k \)[/tex] using the first beam:
For the first beam, we have:
- Width = [tex]\( \frac{1}{3} \)[/tex] foot
- Height = [tex]\( \frac{1}{3} \)[/tex] foot
- Length = 10 feet
- Weight = 20 tons
Plug these values into the formula to solve for [tex]\( k \)[/tex]:
[tex]\[
20 = k \cdot \frac{\frac{1}{3} \cdot \left(\frac{1}{3}\right)^2}{10}
\][/tex]
Simplifying inside the fraction:
[tex]\[
20 = k \cdot \frac{\frac{1}{3} \cdot \frac{1}{9}}{10}
\][/tex]
[tex]\[
20 = k \cdot \frac{1}{27 \cdot 10}
\][/tex]
[tex]\[
20 = k \cdot \frac{1}{270}
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = 20 \times 270 = 5400
\][/tex]
3. Calculate the maximum weight for the second beam:
Now, use the same formula to find the weight the second beam can support with the given dimensions:
- Width = [tex]\( \frac{2}{3} \)[/tex] foot
- Height = [tex]\( \frac{1}{2} \)[/tex] foot
- Length = 18 feet
The formula becomes:
[tex]\[
W = 5400 \cdot \frac{\frac{2}{3} \cdot \left(\frac{1}{2}\right)^2}{18}
\][/tex]
Calculate the fraction:
[tex]\[
W = 5400 \cdot \frac{\frac{2}{3} \cdot \frac{1}{4}}{18}
\][/tex]
[tex]\[
W = 5400 \cdot \frac{\frac{2}{12}}{18}
\][/tex]
[tex]\[
W = 5400 \cdot \frac{1}{108}
\][/tex]
Simplifying:
[tex]\[
W = 50
\][/tex]
Thus, the maximum weight the second beam can support is 50.0 tons.
