To find the equation that relates the width [tex]\( w \)[/tex] and the length [tex]\( l \)[/tex] of a rectangle with a constant area, we need to evaluate the given data to identify the pattern.
### Step-by-Step Analysis:
1. Given Data:
- Width and Length pairs:
- [tex]\( w = 2 \)[/tex], [tex]\( l = 375 \)[/tex]
- [tex]\( w = 4 \)[/tex], [tex]\( l = 187.5 \)[/tex]
- [tex]\( w = 6 \)[/tex], [tex]\( l = 125 \)[/tex]
- [tex]\( w = 6 \)[/tex], [tex]\( l = 93.75 \)[/tex]
2. Calculate Products:
- For each pair, compute the product [tex]\( w \times l \)[/tex]:
- [tex]\( 2 \times 375 = 750 \)[/tex]
- [tex]\( 4 \times 187.5 = 750 \)[/tex]
- [tex]\( 6 \times 125 = 750 \)[/tex]
- [tex]\( 6 \times 93.75 = 562.5 \)[/tex]
3. Identify the Constant [tex]\( k \)[/tex]:
- From the calculations, observe that for the first three pairs, the product [tex]\( w \times l \)[/tex] is 750. However, the fourth pair gives a product of 562.5, indicating a discrepancy. Assuming there's a typo in the data for the last pair, let’s proceed with the assumption based on the consistent results from the first three pairs.
- The constant [tex]\( k \)[/tex] appears to be 750.
4. Derive the Equation:
- The equation of a rectangle with a constant area [tex]\( k \)[/tex] is given by the formula:
[tex]\[
l = \frac{k}{w}
\][/tex]
- Here, [tex]\( k \)[/tex] is the constant area which we've found to be 750.
5. Conclusion:
- The correct equation that can be used to find any corresponding length [tex]\( l \)[/tex] and width [tex]\( w \)[/tex] that fit the pattern [tex]\( \frac{k}{w} \)[/tex], where [tex]\( l \)[/tex] is the length, [tex]\( w \)[/tex] is the width, and [tex]\( k \)[/tex] is a constant, is:
[tex]\[
l = \frac{750}{w}
\][/tex]
### Final Answer:
Given the patterns in the data and the above derivation, the equation that correctly represents the relationship between the width [tex]\( w \)[/tex] and the length [tex]\( l \)[/tex] is:
[tex]\[
l = \frac{k}{w} \quad \text{where } l \text{ is the length, } w \text{ is the width, and } k \text{ is a constant with value 750}.
\][/tex]
