A Rectangle Has An Area Of $\left(x^2-17x+72\right$\] Square Units. Since The Area Of A Rectangle Is Determined Using The Formula, $A=hw$, What Could Be The Length And Width Of The Rectangle?A. Length = $(x-8$\] Units And Width

Introduction

In mathematics, the area of a rectangle is a fundamental concept that is used to calculate the space inside the rectangle. The area of a rectangle is determined using the formula, $A=hw$, where $A$ is the area, $h$ is the height, and $w$ is the width. In this article, we will explore the relationship between the area of a rectangle and its length and width, using a given quadratic expression to represent the area.

The Area of a Rectangle

The area of a rectangle is calculated by multiplying its length and width. This can be represented mathematically as:

$$A = hw$$

where $A$ is the area, $h$ is the height, and $w$ is the width. In this case, the area of the rectangle is given as $\left(x^2-17x+72\right)$ square units.

The Quadratic Expression

The quadratic expression $\left(x^2-17x+72\right)$ can be factored to determine the possible values of the length and width of the rectangle. Factoring the quadratic expression, we get:

$$\left(x^2-17x+72\right) = \left(x-8\right)\left(x-9\right)$$

Finding the Length and Width

Since the area of the rectangle is determined by the product of its length and width, we can set up an equation using the factored form of the quadratic expression:

$$\left(x-8\right)\left(x-9\right) = hw$$

To find the possible values of the length and width, we can equate the factored form of the quadratic expression to the product of the length and width:

$$\left(x-8\right)\left(x-9\right) = \left(x-8\right)w$$

Simplifying the equation, we get:

$$\left(x-9\right) = w$$

Possible Values of the Length and Width

From the simplified equation, we can see that the possible values of the width are $\left(x-9\right)$ units. Since the length and width are interchangeable, we can also consider the possible values of the length to be $\left(x-8\right)$ units.

Conclusion

In conclusion, the length and width of the rectangle can be represented as $\left(x-8\right)$ units and $\left(x-9\right)$ units, respectively. This is based on the factored form of the quadratic expression and the relationship between the area of a rectangle and its length and width.

Discussion

The discussion category for this article is mathematics, as it involves the use of mathematical concepts and formulas to solve a problem. The article provides a step-by-step solution to the problem, using the factored form of the quadratic expression to determine the possible values of the length and width of the rectangle.

Final Thoughts

In this article, we have explored the relationship between the area of a rectangle and its length and width, using a given quadratic expression to represent the area. The article provides a clear and concise solution to the problem, using mathematical concepts and formulas to determine the possible values of the length and width of the rectangle.

Introduction

In our previous article, we explored the relationship between the area of a rectangle and its length and width, using a given quadratic expression to represent the area. In this article, we will answer some frequently asked questions (FAQs) related to the problem.

Q&A

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

A: The formula for the area of a rectangle is $A = hw$, where $A$ is the area, $h$ is the height, and $w$ is the width.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. In this case, the quadratic expression is $\left(x^2-17x+72\right)$, which can be factored as $\left(x-8\right)\left(x-9\right)$.

Q: What are the possible values of the length and width of the rectangle?

A: The possible values of the length and width of the rectangle are $\left(x-8\right)$ units and $\left(x-9\right)$ units, respectively.

Q: How do I determine the possible values of the length and width?

A: To determine the possible values of the length and width, you need to equate the factored form of the quadratic expression to the product of the length and width. In this case, we get $\left(x-8\right)\left(x-9\right) = \left(x-8\right)w$, which simplifies to $\left(x-9\right) = w$.

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

A: The area of a rectangle is determined by the product of its length and width. In this case, the area of the rectangle is $\left(x^2-17x+72\right)$ square units, which can be represented as $\left(x-8\right)\left(x-9\right)$.

Q: How do I use the factored form of the quadratic expression to determine the possible values of the length and width?

A: To use the factored form of the quadratic expression to determine the possible values of the length and width, you need to equate the factored form to the product of the length and width. In this case, we get $\left(x-8\right)\left(x-9\right) = \left(x-8\right)w$, which simplifies to $\left(x-9\right) = w$.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to the problem of finding the length and width of a rectangle given its area. We have provided step-by-step solutions to the problem, using mathematical concepts and formulas to determine the possible values of the length and width of the rectangle.

Final Thoughts

In this article, we have explored the relationship between the area of a rectangle and its length and width, using a given quadratic expression to represent the area. We have also answered some frequently asked questions (FAQs) related to the problem, providing step-by-step solutions to the problem.