The Dimensions Of A Rectangle Are 50 A 3 B 2 \sqrt{50 A^3 B^2} 50 A 3 B 2 And 200 A 3 \sqrt{200 A^3} 200 A 3 . A Student Found The Perimeter As Follows:$[ \begin{aligned} 2 \sqrt{50 A^3 B^2} + 2 \sqrt{200 A^3} & = 2 \cdot 5 A B \sqrt{2 A} + 2 \cdot 10 A \sqrt{2 A}

Mar 12, 2025 by ADMIN 269 views

**The Dimensions of a Rectangle: A Mathematical Exploration**

Introduction

In mathematics, the study of geometric shapes is a fundamental aspect of understanding various mathematical concepts. Rectangles, in particular, are a type of quadrilateral with four right angles and opposite sides of equal length. In this article, we will delve into the dimensions of a rectangle and explore how to calculate its perimeter using mathematical formulas.

The Dimensions of the Rectangle

The dimensions of the rectangle are given as $\sqrt{50 a^3 b^2}$ and $\sqrt{200 a^3}$ . To understand the significance of these dimensions, let's break them down into their constituent parts.

The first dimension, $\sqrt{50 a^3 b^2}$ , can be simplified by factoring out the square root of 50, which is 5√2. This gives us $5 a b \sqrt{2 a}$ .
The second dimension, $\sqrt{200 a^3}$ , can be simplified by factoring out the square root of 200, which is 10√2. This gives us $10 a \sqrt{2 a}$ .

Calculating the Perimeter

The perimeter of a rectangle is the sum of the lengths of all its sides. In this case, the perimeter can be calculated as follows:

{ \begin{aligned} 2 \sqrt{50 a^3 b^2} + 2 \sqrt{200 a^3} & = 2 \cdot 5 a b \sqrt{2 a} + 2 \cdot 10 a \sqrt{2 a} \end{aligned} }

To simplify this expression, we can combine like terms:

{ \begin{aligned} 2 \cdot 5 a b \sqrt{2 a} + 2 \cdot 10 a \sqrt{2 a} & = 10 a b \sqrt{2 a} + 20 a \sqrt{2 a} \end{aligned} }

Now, we can factor out the common term $\sqrt{2 a}$ :

{ \begin{aligned} 10 a b \sqrt{2 a} + 20 a \sqrt{2 a} & = \sqrt{2 a} (10 a b + 20 a) \end{aligned} }

Simplifying further, we get:

{ \begin{aligned} \sqrt{2 a} (10 a b + 20 a) & = \sqrt{2 a} (10 a b + 20 a) \end{aligned} }

Simplifying the Expression

To simplify the expression, we can factor out the common term $\sqrt{2 a}$ :

{ \begin{aligned} \sqrt{2 a} (10 a b + 20 a) & = \sqrt{2 a} (10 a b + 20 a) \end{aligned} }

Now, we can simplify the expression by combining like terms:

{ \begin{aligned} \sqrt{2 a} (10 a b + 20 a) & = \sqrt{2 a} (10 a b + 20 a) \end{aligned} }

Conclusion

In conclusion, the perimeter of the rectangle can be calculated using the formula:

{ \begin{aligned} 2 \sqrt{50 a^3 b^2} + 2 \sqrt{200 a^3} & = 2 \cdot 5 a b \sqrt{2 a} + 2 \cdot 10 a \sqrt{2 a} \end{aligned} }

By simplifying the expression, we can factor out the common term $\sqrt{2 a}$ and combine like terms to get the final result.

Final Answer

The final answer is $\boxed{2 \cdot 5 a b \sqrt{2 a} + 2 \cdot 10 a \sqrt{2 a}}$ .

References

[1] "Mathematics for Dummies". John Wiley & Sons.
[2] "Geometry: A Comprehensive Introduction". McGraw-Hill Education.

Additional Resources

[1] Khan Academy: Geometry
[2] MIT OpenCourseWare: Geometry

Discussion

Introduction

In our previous article, we explored the dimensions of a rectangle and calculated its perimeter using mathematical formulas. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What are the dimensions of the rectangle?

A: The dimensions of the rectangle are $\sqrt{50 a^3 b^2}$ and $\sqrt{200 a^3}$ .

Q: How do I simplify the expression for the perimeter?

A: To simplify the expression, you can factor out the common term $\sqrt{2 a}$ and combine like terms.

Q: What is the final answer for the perimeter?

A: The final answer for the perimeter is $\boxed{2 \cdot 5 a b \sqrt{2 a} + 2 \cdot 10 a \sqrt{2 a}}$ .

Q: Can you explain the concept of perimeter in more detail?

A: The perimeter of a rectangle is the sum of the lengths of all its sides. In this case, the perimeter can be calculated as follows:

{ \begin{aligned} 2 \sqrt{50 a^3 b^2} + 2 \sqrt{200 a^3} & = 2 \cdot 5 a b \sqrt{2 a} + 2 \cdot 10 a \sqrt{2 a} \end{aligned} }

Q: How do I calculate the perimeter of a rectangle with different dimensions?

A: To calculate the perimeter of a rectangle with different dimensions, you can use the formula:

Perimeter = 2(l + w)

where l is the length and w is the width of the rectangle.

Q: What are some real-world applications of calculating the perimeter of a rectangle?

A: Calculating the perimeter of a rectangle has many real-world applications, such as:

Building design and construction
Landscaping and gardening
Architecture and engineering
Interior design and decoration

Q: Can you provide some examples of how to calculate the perimeter of a rectangle?

A: Here are a few examples:

A rectangle with a length of 10 cm and a width of 5 cm has a perimeter of 2(10 + 5) = 30 cm.
A rectangle with a length of 20 m and a width of 15 m has a perimeter of 2(20 + 15) = 70 m.