The Length Of A Rectangle Is 4 More Than The Width. The Area Of The Rectangle Is 60 Square Yards. What Is The Length Of The Rectangle?Let $w$ Be The Width, And $4 + W$ Be The Length.Which Equation Represents The Situation?$\[ W(4

Introduction

In this article, we will delve into a mathematical problem that involves finding the length of a rectangle given its area and the relationship between its length and width. We will use algebraic equations to represent the situation and solve for the unknown variable.

The Problem

The length of a rectangle is 4 more than the width. The area of the rectangle is 60 square yards. We need to find the length of the rectangle.

Let $w$ be the width, and $4 + w$ be the length. We can represent the situation using an equation.

Representing the Situation

The area of a rectangle is given by the formula:

Area = Length × Width

In this case, the area is 60 square yards, and the length is $4 + w$ and the width is $w$. We can write the equation as:

$w(4 + w) = 60$

Simplifying the Equation

We can simplify the equation by distributing the $w$ to the terms inside the parentheses:

$4w + w^2 = 60$

Rearranging the Equation

We can rearrange the equation to put it in standard quadratic form:

$w^2 + 4w - 60 = 0$

Solving the Equation

We can solve the equation using the quadratic formula:

$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 1$, $b = 4$, and $c = -60$. Plugging these values into the formula, we get:

$w = \frac{-4 \pm \sqrt{4^2 - 4(1)(-60)}}{2(1)}$

$w = \frac{-4 \pm \sqrt{16 + 240}}{2}$

$w = \frac{-4 \pm \sqrt{256}}{2}$

$w = \frac{-4 \pm 16}{2}$

This gives us two possible values for $w$:

$w = \frac{-4 + 16}{2} = 6$

$w = \frac{-4 - 16}{2} = -10$

Since the width cannot be negative, we discard the solution $w = -10$.

Finding the Length

Now that we have found the width, we can find the length by substituting the value of $w$ into the expression $4 + w$:

Length = $4 + w$

Length = $4 + 6$

Length = 10

Conclusion

In this article, we used algebraic equations to represent the situation and solve for the unknown variable. We found that the width of the rectangle is 6 yards, and the length is 10 yards.

Final Answer

The length of the rectangle is 10 yards.

Additional Resources

For more information on algebraic equations and quadratic formulas, please see the following resources:

• Algebraic Equations
• Quadratic Formula

Related Problems

• The length of a rectangle is 5 more than the width. The area of the rectangle is 72 square yards. What is the length of the rectangle?
• The length of a rectangle is 3 more than the width. The area of the rectangle is 48 square yards. What is the length of the rectangle?

Introduction

In our previous article, we explored a mathematical problem that involved finding the length of a rectangle given its area and the relationship between its length and width. We used algebraic equations to represent the situation and solve for the unknown variable. In this article, we will provide a Q&A guide to help you better understand the problem and its solution.

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

A: The length of the rectangle is 4 more than the width. This means that if the width is $w$, then the length is $4 + w$.

Q: How do we represent the situation using an equation?

A: We can represent the situation using the equation $w(4 + w) = 60$, where $w$ is the width and $60$ is the area of the rectangle.

Q: How do we simplify the equation?

A: We can simplify the equation by distributing the $w$ to the terms inside the parentheses, resulting in the equation $4w + w^2 = 60$.

Q: How do we rearrange the equation?

A: We can rearrange the equation to put it in standard quadratic form, resulting in the equation $w^2 + 4w - 60 = 0$.

Q: How do we solve the equation?

A: We can solve the equation using the quadratic formula: $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 4$, and $c = -60$.

Q: What are the possible values for $w$?

A: Plugging the values into the quadratic formula, we get two possible values for $w$: $w = 6$ and $w = -10$. Since the width cannot be negative, we discard the solution $w = -10$.

Q: What is the length of the rectangle?

A: Now that we have found the width, we can find the length by substituting the value of $w$ into the expression $4 + w$: Length = $4 + 6$ = 10.

Q: What if the length of the rectangle is 5 more than the width?

A: If the length of the rectangle is 5 more than the width, we can represent the situation using the equation $w(5 + w) = 72$, where $w$ is the width and $72$ is the area of the rectangle. We can simplify and solve the equation using the same techniques as before.

Q: What if the length of the rectangle is 3 more than the width?

A: If the length of the rectangle is 3 more than the width, we can represent the situation using the equation $w(3 + w) = 48$, where $w$ is the width and $48$ is the area of the rectangle. We can simplify and solve the equation using the same techniques as before.

Conclusion

In this Q&A guide, we have provided answers to common questions related to the problem of finding the length of a rectangle given its area and the relationship between its length and width. We have also provided examples of how to solve similar problems.

Additional Resources

For more information on algebraic equations and quadratic formulas, please see the following resources:

• Algebraic Equations
• Quadratic Formula

Related Problems

• The length of a rectangle is 5 more than the width. The area of the rectangle is 72 square yards. What is the length of the rectangle?
• The length of a rectangle is 3 more than the width. The area of the rectangle is 48 square yards. What is the length of the rectangle?

Note: The above related problems are not solved in this article, but they can be solved using the same techniques and methods used in this article.