The Expressions for Length and Width

Oct 13, 2025 by wbrum.org 65 views

Introduction

In mathematics, expressions are used to represent various quantities or values. When it comes to geometry, expressions can be used to represent the dimensions of shapes, such as the length and width of a patio. In this article, we will explore how to find the area of Vanessa's patio using the given expressions for its length and width.

The Expressions for Length and Width

Vanessa uses the expressions ${$ [3x^2 + 5x + 10]$}$ and ${$ [x^2 - 3x - 1]$}$ to represent the length and width of her patio, respectively. These expressions are quadratic equations, which are polynomial equations of degree two.

Length Expression: ${$ [3x^2 + 5x + 10]$}$

The length expression is given by ${$ [3x^2 + 5x + 10]$}$. This expression represents the length of the patio, which is a quadratic function of the variable ${$ x$}$.

Width Expression: ${$ [x^2 - 3x - 1]$}$

The width expression is given by ${$ [x^2 - 3x - 1]$}$. This expression represents the width of the patio, which is also a quadratic function of the variable ${$ x$}$.

Finding the Area of the Patio

To find the area of the patio, we need to multiply the length and width expressions. The area ${$ (A)$}$ of a rectangle is given by the product of its length and width.

Area Expression: ${$ (A) = (3x^2 + 5x + 10)(x^2 - 3x - 1)$}$

To find the area expression, we multiply the length and width expressions:

${$ (A) = (3x^2 + 5x + 10)(x^2 - 3x - 1)$}$

Using the distributive property, we can expand the product:

${$ (A) = 3x^4 - 9x^3 - 3x^2 + 5x^3 - 15x^2 - 5x + 10x^2 - 30x - 10$}$

Combining like terms, we get:

${$ (A) = 3x^4 - 4x^3 - 8x^2 - 35x - 10$}$

Conclusion

In this article, we explored how to find the area of Vanessa's patio using the given expressions for its length and width. We multiplied the length and width expressions to get the area expression, which is a quadratic function of the variable ${$ x$}$. The area expression is given by ${$ (A) = 3x^4 - 4x^3 - 8x^2 - 35x - 10$}$.

Discussion

What are some real-world applications of finding the area of a patio?
How can we use the area expression to solve problems involving the patio?
What are some other ways to find the area of a patio?

Answer Key

A. ${ $3x^4 - 4x^3 - 8x^2 - 35x - 10\$}$

Additional Resources

For more information on quadratic equations, see Quadratic Equations.
For more information on geometry, see Geometry.
For more information on algebra, see Algebra.
Vanessa's Patio Area: A Mathematical Exploration - Q&A =====================================================

Introduction

In our previous article, we explored how to find the area of Vanessa's patio using the given expressions for its length and width. We multiplied the length and width expressions to get the area expression, which is a quadratic function of the variable ${$ x$}$. In this article, we will answer some frequently asked questions about the area of Vanessa's patio.

Q&A

Q: What is the area of Vanessa's patio?

A: The area of Vanessa's patio is given by the expression ${$ (A) = 3x^4 - 4x^3 - 8x^2 - 35x - 10$}$.

Q: How did you find the area expression?

A: We multiplied the length and width expressions to get the area expression. The length expression is ${$ [3x^2 + 5x + 10]$}$ and the width expression is ${$ [x^2 - 3x - 1]$}$.

Q: What are some real-world applications of finding the area of a patio?

A: Finding the area of a patio can be useful in various real-world applications, such as:

Landscaping: When designing a patio, it's essential to know the area to determine the amount of materials needed for landscaping.
Furniture arrangement: Knowing the area of a patio can help you arrange furniture and other objects in a way that maximizes space.
Outdoor events: If you're planning an outdoor event, such as a party or wedding, knowing the area of the patio can help you determine the number of guests that can be accommodated.

Q: How can we use the area expression to solve problems involving the patio?

A: We can use the area expression to solve problems involving the patio by substituting values for the variable ${$ x$}$ and solving for the area.

Q: What are some other ways to find the area of a patio?

A: There are several other ways to find the area of a patio, including:

Using a calculator: You can use a calculator to find the area of a patio by plugging in the values for the length and width.
Using a formula: You can use a formula, such as ${$ A = lw$}$, to find the area of a patio.
Using a graphing calculator: You can use a graphing calculator to visualize the area of a patio and find the maximum or minimum area.

Conclusion

In this article, we answered some frequently asked questions about the area of Vanessa's patio. We discussed how to find the area expression, real-world applications of finding the area of a patio, and other ways to find the area of a patio.

Discussion

What are some other ways to find the area of a patio?
How can we use the area expression to solve problems involving the patio?
What are some real-world applications of finding the area of a patio?

Answer Key

A: ${ $3x^4 - 4x^3 - 8x^2 - 35x - 10\$}$
A: We multiplied the length and width expressions to get the area expression.
A: Landscaping, furniture arrangement, and outdoor events.
A: We can substitute values for the variable ${$ x$}$ and solve for the area.
A: Using a calculator, using a formula, and using a graphing calculator.

Additional Resources

For more information on quadratic equations, see Quadratic Equations.
For more information on geometry, see Geometry.
For more information on algebra, see Algebra.