Vanessa Uses The Expressions { \left(3x^2 + 5x + 10\right)$}$ And { \left(x^2 - 3x - 1\right)$}$ To Represent The Length And Width Of Her Patio. Which Expression Represents The Area { \left(hw^{\prime}\right)$}$ Of
Introduction
In mathematics, expressions are used to represent various quantities or values. When it comes to real-world applications, such as calculating the area of a patio, understanding the expressions used to represent the dimensions is crucial. In this article, we will explore the expressions used by Vanessa to represent the length and width of her patio and determine which one represents the area.
The Expressions for Length and Width
Vanessa uses the expressions {\left(3x^2 + 5x + 10\right)$}$ and {\left(x^2 - 3x - 1\right)$}$ to represent the length and width of her patio, respectively. These expressions are quadratic equations, which are polynomial equations of degree two. The general form of a quadratic equation is {ax^2 + bx + c = 0$}$, where {a$}$, {b$}$, and {c$}$ are constants.
Understanding the Area Formula
The area of a rectangle, such as a patio, is calculated by multiplying the length and width. The formula for the area is {A = lw$}$, where {A$}$ is the area, {l$}$ is the length, and {w$}$ is the width. In this case, the expressions for length and width are {\left(3x^2 + 5x + 10\right)$}$ and {\left(x^2 - 3x - 1\right)$}$, respectively.
Determining the Correct Expression for Area
To determine which expression represents the area, we need to multiply the expressions for length and width. This can be done by multiplying the two expressions together:
{\left(3x^2 + 5x + 10\right)$}$ * {\left(x^2 - 3x - 1\right)$}$
Using the distributive property, we can expand the product:
${3x^2\left(x^2 - 3x - 1\right) + 5x\left(x^2 - 3x - 1\right) + 10\left(x^2 - 3x - 1\right)\$}
Expanding each term, we get:
${3x^4 - 9x^3 - 3x^2 + 5x^3 - 15x^2 - 5x + 10x^2 - 30x - 10\$}
Combining like terms, we get:
${3x^4 - 4x^3 - 8x^2 - 35x - 10\$}
Conclusion
In conclusion, the expression that represents the area of Vanessa's patio is {\left(3x^4 - 4x^3 - 8x^2 - 35x - 10\right)$}$. This expression is obtained by multiplying the expressions for length and width, which are {\left(3x^2 + 5x + 10\right)$}$ and {\left(x^2 - 3x - 1\right)$}$, respectively.
Understanding the Significance of the Area Expression
The area expression {\left(3x^4 - 4x^3 - 8x^2 - 35x - 10\right)$}$ has significant implications for Vanessa's patio design. The area of the patio is a critical factor in determining the amount of space available for outdoor activities, such as seating, dining, or entertainment. By understanding the expression for the area, Vanessa can make informed decisions about the design and layout of her patio.
Real-World Applications of the Area Expression
The area expression {\left(3x^4 - 4x^3 - 8x^2 - 35x - 10\right)$}$ has numerous real-world applications in various fields, including:
- Architecture: The area expression can be used to calculate the area of buildings, bridges, or other structures.
- Engineering: The area expression can be used to calculate the area of mechanical components, such as gears or bearings.
- Landscaping: The area expression can be used to calculate the area of gardens, parks, or other outdoor spaces.
Conclusion
Introduction
In our previous article, we explored the expressions used by Vanessa to represent the length and width of her patio and determined which one represents the area. In this article, we will provide a Q&A guide to help you better understand the area expression and its significance.
Q: What is the area expression for Vanessa's patio?
A: The area expression for Vanessa's patio is {\left(3x^4 - 4x^3 - 8x^2 - 35x - 10\right)$}$.
Q: How was the area expression obtained?
A: The area expression was obtained by multiplying the expressions for length and width, which are {\left(3x^2 + 5x + 10\right)$}$ and {\left(x^2 - 3x - 1\right)$}$, respectively.
Q: What is the significance of the area expression?
A: The area expression has significant implications for Vanessa's patio design. The area of the patio is a critical factor in determining the amount of space available for outdoor activities, such as seating, dining, or entertainment.
Q: What are some real-world applications of the area expression?
A: The area expression has numerous real-world applications in various fields, including:
- Architecture: The area expression can be used to calculate the area of buildings, bridges, or other structures.
- Engineering: The area expression can be used to calculate the area of mechanical components, such as gears or bearings.
- Landscaping: The area expression can be used to calculate the area of gardens, parks, or other outdoor spaces.
Q: How can the area expression be used in real-world scenarios?
A: The area expression can be used in various real-world scenarios, such as:
- Designing a patio: The area expression can be used to calculate the area of a patio and determine the amount of space available for outdoor activities.
- Calculating the area of a building: The area expression can be used to calculate the area of a building and determine the amount of space available for various uses.
- Designing a garden: The area expression can be used to calculate the area of a garden and determine the amount of space available for various plants and features.
Q: What are some common mistakes to avoid when working with the area expression?
A: Some common mistakes to avoid when working with the area expression include:
- Not multiplying the expressions correctly: Make sure to multiply the expressions for length and width correctly to obtain the area expression.
- Not simplifying the expression: Make sure to simplify the expression to obtain the final area expression.
- Not considering the units: Make sure to consider the units of the area expression, such as square meters or square feet.
Q: How can the area expression be used in mathematical problems?
A: The area expression can be used in various mathematical problems, such as:
- Solving quadratic equations: The area expression can be used to solve quadratic equations and determine the values of the variables.
- Calculating the area of a shape: The area expression can be used to calculate the area of a shape, such as a rectangle or a triangle.
- Designing a mathematical model: The area expression can be used to design a mathematical model and determine the values of the variables.
Conclusion
In conclusion, the area expression {\left(3x^4 - 4x^3 - 8x^2 - 35x - 10\right)$}$ has significant implications for Vanessa's patio design and has numerous real-world applications in various fields. By understanding the area expression and its significance, you can better design and calculate the area of various shapes and spaces.