The area of a rectangle is greater than 10. What is a reasonable domain for the shorter side of the rectangle?

To determine a reasonable domain for the shorter side of a rectangle given that the area is greater than 10, we can follow these steps: 1. **Define the variables:** - Let \( w \) be the width (shorter side) of the rectangle. - Let \( l \) be the length (longer side) of the rectangle. 2. **Area of the rectangle:** The area \( A \) of a rectangle is given by the formula: \[ A = w \times l \] We know that the area must be greater than 10: \[ w \times l > 10 \] 3. **Express the length in terms of the width:** Rearranging the inequality gives: \[ l > \frac{10}{w} \] This means that for any given width \( w \), the length \( l \) must be greater than \( \frac{10}{w} \). 4. **Determine the domain for \( w \):** - Since \( w \) represents a physical dimension (the width of the rectangle), it must be positive: \[ w > 0 \] - Additionally, as \( w \) approaches 0, \( l \) would need to be infinitely large to satisfy the area condition, which is not practical. Therefore, we need to consider a reasonable upper limit for \( w \). 5. **Conclusion:** The domain for the shorter side \( w \) must be: \[ w > 0 \] There is no strict upper limit for \( w \) in this context, but it should be reasonable based on practical dimensions of a rectangle. Thus, a reasonable domain for the shorter side of the rectangle is: \[ (0, \infty) \]