Understanding the Original Patio

Oct 13, 2025 by talynt.org 61 views

Introduction

Maria has a square brick patio that she wants to renovate by reducing the width by 3 feet and increasing the length by 3 feet. In this article, we will explore how to write expressions for the length and width of the new patio and then find the area of the new patio.

Understanding the Original Patio

The original patio is a square with each side having a length of $x$ feet. Since it is a square, all sides are equal in length. We can represent the length and width of the original patio as:

Length: $x$
Width: $x$

Writing Expressions for the New Patio

To find the expressions for the length and width of the new patio, we need to apply the changes that Maria wants to make. She wants to reduce the width by 3 feet, so the new width will be $x - 3$ . She also wants to increase the length by 3 feet, so the new length will be $x + 3$ .

Therefore, the expressions for the length and width of the new patio are:

Length: $x + 3$
Width: $x - 3$

Finding the Area of the New Patio

The area of a rectangle (such as the new patio) is given by the formula:

Area = Length × Width

Substituting the expressions for the length and width of the new patio, we get:

Area = $(x + 3)(x - 3)$

To find the area, we need to multiply the two binomials using the FOIL method:

FOIL stands for First, Outer, Inner, Last. It is a technique used to multiply two binomials.

First: $x \times x = x^2$
Outer: $x \times -3 = -3x$
Inner: $3 \times x = 3x$
Last: $3 \times -3 = -9$

Now, we add the terms:

Area = $x^2 - 3x + 3x - 9$

Simplifying the expression, we get:

Area = $x^2 - 9$

Conclusion

In this article, we have written expressions for the length and width of Maria's new patio and found the area of the new patio. We have used the FOIL method to multiply the binomials and simplified the expression to find the area.

Key Takeaways

The expressions for the length and width of the new patio are $x + 3$ and $x - 3$ , respectively.
The area of the new patio is given by the expression $x^2 - 9$ .

Real-World Applications

This problem has real-world applications in architecture, engineering, and design. When designing a patio or a building, it is essential to consider the dimensions and area of the space to ensure that it meets the needs of the users.

Future Directions

In the future, we can explore more complex problems involving geometry and algebra. We can also apply these concepts to real-world problems in fields such as physics, engineering, and computer science.

References

[1] Khan Academy. (n.d.). Algebra. Retrieved from https://www.khanacademy.org/math/algebra
[2] Math Open Reference. (n.d.). Algebra. Retrieved from https://www.mathopenref.com/algebra.html

Glossary

FOIL method: A technique used to multiply two binomials.
Binomial: An algebraic expression consisting of two terms.
Area: The amount of space inside a two-dimensional shape.
Length: A measure of the distance between two points.
Width: A measure of the distance between two points in a direction perpendicular to the length.
Maria's Patio Renovation: A Mathematical Approach - Q&A =====================================================

Introduction

In our previous article, we explored how to write expressions for the length and width of Maria's new patio and found the area of the new patio. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the original length and width of the patio?

A: The original length and width of the patio are both $x$ feet.

Q: How do we find the expressions for the length and width of the new patio?

A: To find the expressions for the length and width of the new patio, we need to apply the changes that Maria wants to make. She wants to reduce the width by 3 feet, so the new width will be $x - 3$ . She also wants to increase the length by 3 feet, so the new length will be $x + 3$ .

Q: What is the area of the new patio?

A: The area of the new patio is given by the expression $x^2 - 9$ .

Q: How do we multiply the binomials to find the area?

A: We use the FOIL method to multiply the binomials. The FOIL method stands for First, Outer, Inner, Last. It is a technique used to multiply two binomials.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomials. It involves multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q: Can we use the FOIL method to multiply any two binomials?

A: Yes, we can use the FOIL method to multiply any two binomials.

Q: What is the difference between the original patio and the new patio?

A: The original patio is a square with each side having a length of $x$ feet. The new patio is also a rectangle with a length of $x + 3$ feet and a width of $x - 3$ feet.

Q: How do we find the area of a rectangle?

A: The area of a rectangle is given by the formula: Area = Length × Width.

Q: Can we use the same formula to find the area of a square?

A: Yes, we can use the same formula to find the area of a square.

Q: What is the relationship between the length and width of a rectangle?

A: The length and width of a rectangle are perpendicular to each other.

Q: Can we use the same formula to find the area of a triangle?

A: No, we cannot use the same formula to find the area of a triangle. The formula for the area of a triangle is: Area = (base × height) / 2.