Equations:The Rectangle's Length Is 4 More Than Twice Its Width. Its Area Is 240 Square Centimeters.Complete The Work To Find The Dimensions Of The Rectangle.$ \begin{array}{l} w(2w + 4) = 240 \\ 2w^2 + 4w = 240 \\ 2w^2 + 4w - 240 = 0 \\ 2(w +
Introduction
In this article, we will delve into the world of mathematics and solve a problem involving a rectangle. The problem states that the rectangle's length is 4 more than twice its width, and its area is 240 square centimeters. We will use algebraic equations to find the dimensions of the rectangle.
Understanding the Problem
Let's break down the problem and understand what is being asked. We have a rectangle with a width of w centimeters. The length of the rectangle is 4 more than twice its width, which can be expressed as 2w + 4 centimeters. The area of the rectangle is given as 240 square centimeters.
Setting Up the Equation
We can set up an equation to represent the area of the rectangle. The area of a rectangle is given by the formula length × width. In this case, the length is 2w + 4 and the width is w. So, we can write the equation as:
w(2w + 4) = 240
Expanding and Simplifying the Equation
To solve the equation, we need to expand and simplify it. We can start by distributing the w to the terms inside the parentheses:
2w^2 + 4w = 240
Rearranging the Equation
Next, we can rearrange the equation to set it equal to zero. This will help us solve for the value of w:
2w^2 + 4w - 240 = 0
Solving the Quadratic Equation
We now have a quadratic equation in the form ax^2 + bx + c = 0, where a = 2, b = 4, and c = -240. We can solve this equation using the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the Values
We can plug in the values of a, b, and c into the quadratic formula:
w = (-(4) ± √((4)^2 - 4(2)(-240))) / 2(2)
Simplifying the Expression
We can simplify the expression inside the square root:
w = (-4 ± √(16 + 1920)) / 4
w = (-4 ± √1936) / 4
w = (-4 ± 44) / 4
Finding the Values of w
We now have two possible values for w:
w = (-4 + 44) / 4 or w = (-4 - 44) / 4
w = 40 / 4 or w = -48 / 4
w = 10 or w = -12
Ignoring the Negative Value
Since the width of a rectangle cannot be negative, we can ignore the negative value of w. Therefore, the width of the rectangle is w = 10 centimeters.
Finding the Length
Now that we have the width, we can find the length of the rectangle. The length is given by the expression 2w + 4, so we can plug in the value of w:
length = 2(10) + 4
length = 20 + 4
length = 24
Conclusion
In this article, we used algebraic equations to find the dimensions of a rectangle. We started with the equation w(2w + 4) = 240 and expanded and simplified it to get the quadratic equation 2w^2 + 4w - 240 = 0. We then solved the quadratic equation using the quadratic formula and found the value of w to be 10 centimeters. Finally, we used the value of w to find the length of the rectangle, which was 24 centimeters.
Final Answer
Q: What is the main concept behind solving the rectangle's dimensions?
A: The main concept behind solving the rectangle's dimensions is to use algebraic equations to find the values of the width and length of the rectangle.
Q: What is the formula for the area of a rectangle?
A: The formula for the area of a rectangle is length × width.
Q: How do we set up the equation for the area of the rectangle?
A: We set up the equation by using the formula for the area of a rectangle and substituting the given values. In this case, the length is 2w + 4 and the width is w, so the equation becomes w(2w + 4) = 240.
Q: What is the quadratic formula, and how do we use it to solve the equation?
A: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation. We use the quadratic formula to solve the equation by plugging in the values of a, b, and c and simplifying the expression.
Q: What is the significance of the quadratic formula in solving the equation?
A: The quadratic formula is significant because it allows us to solve quadratic equations that cannot be factored easily. In this case, the quadratic equation 2w^2 + 4w - 240 = 0 cannot be factored easily, so we use the quadratic formula to solve it.
Q: What is the final answer to the problem, and how do we interpret it?
A: The final answer to the problem is w = 10 centimeters and length = 24 centimeters. This means that the width of the rectangle is 10 centimeters and the length is 24 centimeters.
Q: What are some common mistakes to avoid when solving the rectangle's dimensions?
A: Some common mistakes to avoid when solving the rectangle's dimensions include:
- Not setting up the equation correctly
- Not simplifying the expression correctly
- Not using the quadratic formula correctly
- Not interpreting the final answer correctly
Q: What are some real-world applications of solving the rectangle's dimensions?
A: Some real-world applications of solving the rectangle's dimensions include:
- Designing buildings and structures
- Creating art and graphics
- Solving problems in physics and engineering
- Understanding geometric shapes and patterns
Q: How can we use the skills learned from solving the rectangle's dimensions in other areas of mathematics?
A: We can use the skills learned from solving the rectangle's dimensions in other areas of mathematics, such as:
- Solving quadratic equations
- Understanding geometric shapes and patterns
- Applying algebraic concepts to real-world problems
- Developing problem-solving skills and critical thinking.