The Area Of A Rectangle Is $66 \, \text{yd}^2$, And The Length Of The Rectangle Is 7 Yd Less Than Three Times The Width. Find The Dimensions Of The Rectangle.

Introduction

In this article, we will explore the problem of finding the dimensions of a rectangle given its area and a relationship between its length and width. The area of a rectangle is given by the formula $A = lw$, where $A$ is the area, $l$ is the length, and $w$ is the width. We will use this formula to find the dimensions of the rectangle.

Problem Statement

The area of a rectangle is $66 \, \text{yd}^2$, and the length of the rectangle is 7 yd less than three times the width. Let's denote the width as $w$ and the length as $l$. We can write the relationship between the length and the width as $l = 3w - 7$.

Formulating the Equation

We know that the area of the rectangle is $66 \, \text{yd}^2$, so we can write the equation $lw = 66$. Substituting the expression for the length $l = 3w - 7$ into this equation, we get $(3w - 7)w = 66$.

Solving the Equation

Expanding the left-hand side of the equation, we get $3w^2 - 7w = 66$. Rearranging the terms, we get $3w^2 - 7w - 66 = 0$. This is a quadratic equation in $w$.

Using the Quadratic Formula

To solve the quadratic equation $3w^2 - 7w - 66 = 0$, we can use the quadratic formula: $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 3$, $b = -7$, and $c = -66$. Plugging these values into the formula, we get $w = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(3)(-66)}}{2(3)}$.

Simplifying the Expression

Simplifying the expression under the square root, we get $w = \frac{7 \pm \sqrt{49 + 792}}{6}$. This simplifies to $w = \frac{7 \pm \sqrt{841}}{6}$.

Finding the Value of $w$

The square root of 841 is 29, so we can simplify the expression to $w = \frac{7 \pm 29}{6}$. This gives us two possible values for $w$: $w = \frac{7 + 29}{6}$ and $w = \frac{7 - 29}{6}$.

Calculating the Values of $w$

Calculating the values of $w$, we get $w = \frac{36}{6}$ and $w = \frac{-22}{6}$. Simplifying these expressions, we get $w = 6$ and $w = -\frac{11}{3}$.

Finding the Value of $l$

Since the width $w$ cannot be negative, we discard the solution $w = -\frac{11}{3}$. To find the value of the length $l$, we substitute the value of $w$ into the expression $l = 3w - 7$. This gives us $l = 3(6) - 7$.

Calculating the Value of $l$

Calculating the value of $l$, we get $l = 18 - 7$. This simplifies to $l = 11$.

Conclusion

In this article, we found the dimensions of a rectangle given its area and a relationship between its length and width. The width of the rectangle is 6 yd, and the length is 11 yd.

The Final Answer

Introduction

In our previous article, we explored the problem of finding the dimensions of a rectangle given its area and a relationship between its length and width. In this article, we will answer some common questions related to the problem.

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

A: The formula for the area of a rectangle is $A = lw$, where $A$ is the area, $l$ is the length, and $w$ is the width.

Q: How do I find the dimensions of a rectangle given its area and a relationship between its length and width?

A: To find the dimensions of a rectangle given its area and a relationship between its length and width, you can use the formula $A = lw$ and substitute the expression for the length $l$ into the equation. This will give you a quadratic equation in $w$, which you can solve using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I simplify the expression under the square root in the quadratic formula?

A: To simplify the expression under the square root in the quadratic formula, you can first simplify the expression inside the square root by combining like terms. Then, you can factor the expression inside the square root, if possible, or use the square root property to simplify the expression.

Q: What is the square root property?

A: The square root property states that if $x^2 = k$, then $x = \pm \sqrt{k}$.

Q: How do I find the value of $l$ given the value of $w$?

A: To find the value of $l$ given the value of $w$, you can substitute the value of $w$ into the expression $l = 3w - 7$.

Q: What if the value of $w$ is negative?

A: If the value of $w$ is negative, it means that the solution is extraneous and should be discarded.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. However, you will need to use the complex conjugate property to simplify the expression.

Q: What is the complex conjugate property?

A: The complex conjugate property states that if $a + bi$ is a complex number, then its complex conjugate is $a - bi$.

Conclusion

In this article, we answered some common questions related to finding the dimensions of a rectangle given its area and a relationship between its length and width. We hope that this article has been helpful in clarifying any confusion you may have had.

The Final Answer

The final answer is: $\boxed{6, 11}$