Equating the Area to 250 Square Feet

Oct 13, 2025 by talynt.org 95 views

In this problem, we are given a rectangular park with an area of 250 square feet. The length of the park is 7 feet more than twice the width, denoted as ${$ w$}$. Our goal is to find the equations that model this situation in terms of ${$ w$}$.

Defining the Variables

Let's start by defining the variables involved in this problem. We have the width of the park, denoted as ${$ w$}$, and the length of the park, which is 7 feet more than twice the width. We can represent the length as ${ $2w + 7\$}$ .

Calculating the Area

The area of a rectangle is given by the formula: Area = Length × Width. In this case, the area is 250 square feet, and we want to express the length in terms of the width. So, we can write the equation as:

Area = (2w + 7) × w

Simplifying the Equation

Now, let's simplify the equation by multiplying the terms:

Area = 2w^2 + 7w

Equating the Area to 250 Square Feet

Since the area is given as 250 square feet, we can set up the equation:

2w^2 + 7w = 250

Rearranging the Equation

To make it easier to solve, let's rearrange the equation by subtracting 250 from both sides:

2w^2 + 7w - 250 = 0

Solving the Quadratic Equation

This is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 2, b = 7, and c = -250. We can solve this equation using the quadratic formula:

w = (-b ± √(b^2 - 4ac)) / 2a

Substituting the Values

Now, let's substitute the values of a, b, and c into the quadratic formula:

w = (-(7) ± √((7)^2 - 4(2)(-250))) / 2(2)

Simplifying the Expression

Simplifying the expression under the square root, we get:

w = (-7 ± √(49 + 2000)) / 4

w = (-7 ± √2049) / 4

Solving for w

Now, let's solve for w by simplifying the expression:

w = (-7 ± 45.03) / 4

We have two possible solutions for w:

w = (-7 + 45.03) / 4 or w = (-7 - 45.03) / 4

w = 38.03 / 4 or w = -52.03 / 4

w = 9.51 or w = -13.01

Checking the Solutions

Since the width of the park cannot be negative, we can discard the solution w = -13.01. Therefore, the only valid solution is w = 9.51.

Conclusion

In conclusion, the equation that models this situation is:

2w^2 + 7w - 250 = 0

This equation represents the relationship between the width and the area of the rectangular park. By solving this equation, we can find the value of the width, which is w = 9.51 feet.

Answer

The correct answer is:

$A. \[$w(2w + 7)\$<br/> **Frequently Asked Questions (FAQs) about the Rectangular Park Problem**$

In this article, we will address some common questions and concerns related to the rectangular park problem. If you have any further questions or need clarification on any of the topics, feel free to ask.

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

A: The formula for the area of a rectangle is: Area = Length × Width.

Q: How do I calculate the length of the park in terms of the width?

A: To calculate the length of the park in terms of the width, you can use the formula: Length = 2w + 7, where w is the width of the park.

Q: What is the equation that models the situation in terms of w?

A: The equation that models the situation in terms of w is: 2w^2 + 7w - 250 = 0.

Q: How do I solve the quadratic equation 2w^2 + 7w - 250 = 0?

A: To solve the quadratic equation 2w^2 + 7w - 250 = 0, you can use the quadratic formula: w = (-b ± √(b^2 - 4ac)) / 2a, where a = 2, b = 7, and c = -250.

Q: What are the possible solutions for w?

A: The possible solutions for w are w = (-7 + 45.03) / 4 and w = (-7 - 45.03) / 4. However, since the width of the park cannot be negative, we can discard the solution w = -13.01, and the only valid solution is w = 9.51.

Q: What is the value of the width of the park?

A: The value of the width of the park is w = 9.51 feet.

Q: What is the relationship between the width and the area of the rectangular park?

A: The relationship between the width and the area of the rectangular park is given by the equation: 2w^2 + 7w - 250 = 0.

Q: How do I check if the solution is valid?

A: To check if the solution is valid, you can plug the value of w back into the original equation and verify that it satisfies the equation.

Q: What are some common mistakes to avoid when solving the quadratic equation?

A: Some common mistakes to avoid when solving the quadratic equation include:

Not simplifying the expression under the square root
Not checking if the solution is valid
Not using the correct values for a, b, and c in the quadratic formula

Conclusion

In conclusion, the rectangular park problem is a classic example of a quadratic equation. By understanding the formula for the area of a rectangle, calculating the length of the park in terms of the width, and solving the quadratic equation, we can find the value of the width of the park. We hope this article has been helpful in addressing some common questions and concerns related to the rectangular park problem. If you have any further questions or need clarification on any of the topics, feel free to ask.