The Area, $A$, Of A Rectangle Is $120x^2 + 78x - 90$, And The Length, $l$, Of The Rectangle Is $12x + 15$. Which Of The Following Gives The Width, $w$, Of The Rectangle?A. $9x + 4$ B. $10x -

Introduction

In mathematics, the area and dimensions of a rectangle are fundamental concepts that are used to solve various problems. Given the area and length of a rectangle, we can find its width. In this article, we will explore how to find the width of a rectangle when its area and length are given.

The Area of a Rectangle

The area of a rectangle is given by the formula:

$$A = l \times w$$

where $A$ is the area, $l$ is the length, and $w$ is the width.

The Given Area and Length

The area of the rectangle is given as $120x^2 + 78x - 90$, and the length is given as $12x + 15$. We need to find the width, $w$, of the rectangle.

Finding the Width

To find the width, we can use the formula:

$$A = l \times w$$

Substituting the given values, we get:

$$120x^2 + 78x - 90 = (12x + 15) \times w$$

Solving for the Width

To solve for the width, we can divide both sides of the equation by the length:

$$w = \frac{120x^2 + 78x - 90}{12x + 15}$$

Simplifying the Expression

To simplify the expression, we can factor the numerator:

$$w = \frac{(12x + 15)(10x - 6)}{12x + 15}$$

Canceling Out the Common Factor

We can cancel out the common factor $(12x + 15)$:

$$w = 10x - 6$$

Conclusion

Therefore, the width of the rectangle is given by the expression $10x - 6$. This is the correct answer.

Comparison with the Given Options

Let's compare our answer with the given options:

A. $9x + 4$ B. $10x - 6$

Our answer matches option B.

Final Answer

The final answer is $\boxed{10x - 6}$.

Introduction

In our previous article, we explored how to find the width of a rectangle when its area and length are given. In this article, we will answer some frequently asked questions related to the area and dimensions of a rectangle.

Q: What is the formula for the area of a rectangle?

A: The formula for the area of a rectangle is:

$$A = l \times w$$

where $A$ is the area, $l$ is the length, and $w$ is the width.

Q: How do I find the width of a rectangle when its area and length are given?

A: To find the width, you can use the formula:

$$A = l \times w$$

Substitute the given values and solve for the width.

Q: What if the area and length are given as expressions involving variables?

A: In that case, you can use algebraic techniques to solve for the width. For example, if the area is given as $120x^2 + 78x - 90$ and the length is given as $12x + 15$, you can substitute these values into the formula and solve for the width.

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

A: To simplify the expression, you can factor the numerator and cancel out any common factors.

Q: What if I get a negative value for the width?

A: If you get a negative value for the width, it means that the rectangle is not possible with the given area and length. In this case, you need to recheck your calculations or consider a different solution.

Q: Can I use the same method to find the length of a rectangle when its area and width are given?

A: Yes, you can use the same method to find the length of a rectangle when its area and width are given. Simply substitute the given values into the formula and solve for the length.

Q: What if the area and width are given as expressions involving variables?

A: In that case, you can use algebraic techniques to solve for the length. For example, if the area is given as $120x^2 + 78x - 90$ and the width is given as $10x - 6$, you can substitute these values into the formula and solve for the length.

Q: How do I know if the rectangle is possible with the given area and dimensions?

A: To determine if the rectangle is possible, you need to check if the area and dimensions satisfy the conditions of a rectangle. For example, the length and width must be positive, and the area must be equal to the product of the length and width.

Q: Can I use this method to find the dimensions of other shapes, such as triangles or circles?

A: No, this method is specifically designed for finding the dimensions of rectangles. For other shapes, you need to use different formulas and techniques.

Q: Where can I learn more about the area and dimensions of rectangles?

A: You can learn more about the area and dimensions of rectangles by consulting math textbooks, online resources, or seeking help from a math teacher or tutor.

Conclusion

In this article, we have answered some frequently asked questions related to the area and dimensions of rectangles. We hope that this Q&A article has provided you with a better understanding of the topic and has helped you to solve problems involving the area and dimensions of rectangles.