The Perimeter Of A Rectangle Is 34 M, And Its Area Is 60 M². Find The Length And Width Of The Rectangle.
Introduction
In mathematics, the perimeter and area of a rectangle are two fundamental concepts that are often used to describe the shape and size of a rectangle. The perimeter of a rectangle is the total distance around its edges, while the area is the amount of space inside the rectangle. In this article, we will explore the relationship between the perimeter and area of a rectangle, and use this knowledge to find the length and width of a rectangle with a given perimeter and area.
The Perimeter of a Rectangle
The perimeter of a rectangle is calculated by adding up the lengths of all its sides. Since a rectangle has two pairs of equal sides, the perimeter can be calculated using the formula:
P = 2(l + w)
where P is the perimeter, l is the length, and w is the width.
In our problem, the perimeter of the rectangle is given as 34 m. We can use this information to write an equation:
2(l + w) = 34
The Area of a Rectangle
The area of a rectangle is calculated by multiplying the length and width of the rectangle. The formula for the area of a rectangle is:
A = lw
where A is the area, l is the length, and w is the width.
In our problem, the area of the rectangle is given as 60 m². We can use this information to write an equation:
lw = 60
Solving the Equations
Now that we have two equations, we can use them to solve for the length and width of the rectangle. We can start by solving the first equation for l + w:
l + w = 34/2 l + w = 17
Next, we can substitute this expression into the second equation:
lw = 60 (l + w)(l - w) = 60 l² - w² = 60
Using the Difference of Squares Formula
The equation l² - w² = 60 can be factored using the difference of squares formula:
a² - b² = (a + b)(a - b)
In this case, we can let a = l and b = w. Then, we can factor the equation as:
(l + w)(l - w) = 60
Substituting the Value of l + w
We know that l + w = 17, so we can substitute this value into the equation:
(17)(l - w) = 60
Solving for l - w
Now, we can solve for l - w by dividing both sides of the equation by 17:
l - w = 60/17 l - w = 3.53
Finding the Length and Width
Now that we have two equations, we can solve for the length and width of the rectangle. We can start by adding the two equations:
l + w = 17 l - w = 3.53
Adding the Equations
When we add the two equations, we get:
2l = 20.53
Solving for l
Now, we can solve for l by dividing both sides of the equation by 2:
l = 20.53/2 l = 10.265
Finding the Width
Now that we have the length, we can find the width by substituting the value of l into one of the original equations. We can use the equation l + w = 17:
10.265 + w = 17
Solving for w
Now, we can solve for w by subtracting 10.265 from both sides of the equation:
w = 17 - 10.265 w = 6.735
Conclusion
In this article, we used the perimeter and area of a rectangle to find the length and width of a rectangle with a given perimeter and area. We started by writing two equations based on the perimeter and area, and then used algebraic techniques to solve for the length and width. We found that the length of the rectangle is approximately 10.265 m, and the width is approximately 6.735 m.
The Importance of Perimeter and Area
The perimeter and area of a rectangle are two fundamental concepts in mathematics that are used to describe the shape and size of a rectangle. The perimeter is the total distance around the edges of the rectangle, while the area is the amount of space inside the rectangle. Understanding the relationship between the perimeter and area of a rectangle is crucial in many real-world applications, such as architecture, engineering, and design.
Real-World Applications
The perimeter and area of a rectangle have many real-world applications. For example, in architecture, the perimeter of a building is used to determine the amount of materials needed for construction. In engineering, the area of a rectangle is used to calculate the amount of space needed for a particular design. In design, the perimeter and area of a rectangle are used to create aesthetically pleasing and functional designs.
Conclusion
In conclusion, the perimeter and area of a rectangle are two fundamental concepts in mathematics that are used to describe the shape and size of a rectangle. Understanding the relationship between the perimeter and area of a rectangle is crucial in many real-world applications. By using algebraic techniques, we can solve for the length and width of a rectangle with a given perimeter and area. We hope that this article has provided a clear and concise explanation of the perimeter and area of a rectangle, and has inspired readers to explore the many real-world applications of these concepts.
Introduction
In our previous article, we explored the relationship between the perimeter and area of a rectangle, and used this knowledge to find the length and width of a rectangle with a given perimeter and area. In this article, we will answer some of the most frequently asked questions about the perimeter and area of a rectangle.
Q: What is the perimeter of a rectangle?
A: The perimeter of a rectangle is the total distance around its edges. It is calculated by adding up the lengths of all its sides. The formula for the perimeter of a rectangle is:
P = 2(l + w)
where P is the perimeter, l is the length, and w is the width.
Q: What is the area of a rectangle?
A: The area of a rectangle is the amount of space inside the rectangle. It is calculated by multiplying the length and width of the rectangle. The formula for the area of a rectangle is:
A = lw
where A is the area, l is the length, and w is the width.
Q: How do I find the length and width of a rectangle with a given perimeter and area?
A: To find the length and width of a rectangle with a given perimeter and area, you can use the following steps:
- Write an equation based on the perimeter: P = 2(l + w)
- Write an equation based on the area: A = lw
- Solve the two equations simultaneously to find the length and width.
Q: What is the difference between the perimeter and area of a rectangle?
A: The perimeter of a rectangle is the total distance around its edges, while the area is the amount of space inside the rectangle. The perimeter is a measure of the shape of the rectangle, while the area is a measure of the size of the rectangle.
Q: How do I calculate the perimeter and area of a rectangle with a given length and width?
A: To calculate the perimeter and area of a rectangle with a given length and width, you can use the following formulas:
P = 2(l + w) A = lw
Q: What are some real-world applications of the perimeter and area of a rectangle?
A: The perimeter and area of a rectangle have many real-world applications, such as:
- Architecture: The perimeter of a building is used to determine the amount of materials needed for construction.
- Engineering: The area of a rectangle is used to calculate the amount of space needed for a particular design.
- Design: The perimeter and area of a rectangle are used to create aesthetically pleasing and functional designs.
Q: How do I use the perimeter and area of a rectangle to solve problems in real-world applications?
A: To use the perimeter and area of a rectangle to solve problems in real-world applications, you can follow these steps:
- Identify the problem and the given information.
- Write an equation based on the perimeter or area.
- Solve the equation to find the unknown value.
- Use the solution to solve the problem.
Q: What are some common mistakes to avoid when working with the perimeter and area of a rectangle?
A: Some common mistakes to avoid when working with the perimeter and area of a rectangle include:
- Confusing the perimeter and area formulas.
- Not using the correct units of measurement.
- Not checking the solution for reasonableness.
Conclusion
In this article, we have answered some of the most frequently asked questions about the perimeter and area of a rectangle. We hope that this article has provided a clear and concise explanation of the perimeter and area of a rectangle, and has inspired readers to explore the many real-world applications of these concepts.