The Area Of A Rectangle Is Greater Than 10. What Is A Reasonable Domain For The Shorter Side Of The Rectangle?A. $x \ \textgreater \ 0$ B. $x \ \textgreater \ 5$ C. $x \ \textgreater \ \sqrt{10}$ D. $x \

Introduction

In mathematics, the area of a rectangle is calculated by multiplying its length and width. Given that the area of a rectangle is greater than 10, we need to determine a reasonable domain for the shorter side of the rectangle. This problem involves understanding the relationship between the area, length, and width of a rectangle and applying mathematical concepts to find a suitable domain for the shorter side.

Understanding the Problem

Let's denote the length of the rectangle as $l$ and the width as $w$. The area of the rectangle, $A$, is given by the formula:

$$A = l \times w$$

We are told that the area of the rectangle is greater than 10, so we can write:

$$A > 10$$

Substituting the formula for the area, we get:

$$l \times w > 10$$

Analyzing the Inequality

To find a reasonable domain for the shorter side of the rectangle, we need to analyze the inequality $l \times w > 10$. Since the area is greater than 10, both the length and width must be greater than 0. However, we are interested in the shorter side, so we need to consider the minimum value of $w$.

Minimum Value of the Shorter Side

Let's assume that the length of the rectangle is fixed at $l = 1$. In this case, the inequality becomes:

$$1 \times w > 10$$

Simplifying the inequality, we get:

$$w > 10$$

However, this is not a reasonable domain for the shorter side, as it would require the width to be greater than 10, which is not possible.

A More Realistic Approach

Let's consider a more realistic approach by assuming that the length of the rectangle is greater than 1. In this case, the inequality becomes:

$$l \times w > 10$$

Since the length is greater than 1, we can divide both sides of the inequality by $l$ to get:

$$w > \frac{10}{l}$$

Finding a Reasonable Domain

To find a reasonable domain for the shorter side, we need to find a value of $l$ that makes the inequality $w > \frac{10}{l}$ true. Let's consider the following options:

• $l = 1$: In this case, the inequality becomes $w > 10$, which is not a reasonable domain for the shorter side.
• $l = 2$: In this case, the inequality becomes $w > 5$, which is a more reasonable domain for the shorter side.
• $l = 3$: In this case, the inequality becomes $w > \frac{10}{3}$, which is also a reasonable domain for the shorter side.

Conclusion

Based on our analysis, a reasonable domain for the shorter side of the rectangle is $x > 5$. This domain is more realistic and takes into account the relationship between the area, length, and width of the rectangle.

Answer

The correct answer is:

• B. $x > 5$

Discussion

This problem requires a deep understanding of the relationship between the area, length, and width of a rectangle. It also involves applying mathematical concepts to find a suitable domain for the shorter side. The correct answer, $x > 5$, is a reasonable domain that takes into account the given conditions.

Additional Information

• The area of a rectangle is calculated by multiplying its length and width.
• The length and width of a rectangle must be greater than 0.
• The area of a rectangle is greater than 10.
• A reasonable domain for the shorter side of the rectangle is $x > 5$.
  The Area of a Rectangle: A Reasonable Domain for the Shorter Side - Q&A ====================================================================

Introduction

In our previous article, we discussed the problem of finding a reasonable domain for the shorter side of a rectangle given that its area is greater than 10. We analyzed the inequality $l \times w > 10$ and found that a reasonable domain for the shorter side is $x > 5$. In this article, we will provide a Q&A section to further clarify the concepts and provide additional information.

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is $A = l \times w$, where $A$ is the area, $l$ is the length, and $w$ is the width.

Q: Why must the length and width of a rectangle be greater than 0?

A: The length and width of a rectangle must be greater than 0 because a rectangle with a length or width of 0 would not be a rectangle. A rectangle by definition has two pairs of opposite sides of equal length, and all sides must have a positive length.

Q: How do we find a reasonable domain for the shorter side of the rectangle?

A: To find a reasonable domain for the shorter side of the rectangle, we need to analyze the inequality $l \times w > 10$. We can divide both sides of the inequality by $l$ to get $w > \frac{10}{l}$. This gives us a range of values for the shorter side that satisfy the inequality.

Q: What is the minimum value of the shorter side?

A: The minimum value of the shorter side depends on the value of the length. If the length is fixed at $l = 1$, then the minimum value of the shorter side is $w > 10$. However, this is not a reasonable domain for the shorter side. A more realistic approach is to assume that the length is greater than 1, in which case the minimum value of the shorter side is $w > \frac{10}{l}$.

Q: How do we choose the correct domain for the shorter side?

A: To choose the correct domain for the shorter side, we need to consider the given conditions and the relationship between the area, length, and width of the rectangle. In this case, we are given that the area of the rectangle is greater than 10, so we need to find a domain for the shorter side that satisfies this condition.

Q: What is the correct answer?

A: The correct answer is $x > 5$. This domain is a reasonable choice for the shorter side because it takes into account the given conditions and the relationship between the area, length, and width of the rectangle.

Q: What are some additional tips for solving this problem?

A: Some additional tips for solving this problem include:

• Make sure to read the problem carefully and understand the given conditions.
• Use mathematical concepts and formulas to analyze the problem.
• Consider different scenarios and assumptions to find a reasonable domain for the shorter side.
• Check your answer to make sure it satisfies the given conditions.

Conclusion

In this Q&A article, we provided additional information and clarification on the problem of finding a reasonable domain for the shorter side of a rectangle given that its area is greater than 10. We hope this article has been helpful in understanding the concepts and solving the problem.