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

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UpStudy AI · Accepted Answer

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:
$$
(0, \infty)
$$
A reasonable domain for the shorter side of the rectangle is any positive value greater than 0.

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

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