Parking Lot Expansion Solving A Real-World Math Problem
When dealing with geometric expansions, mathematical principles provide us with the tools to solve real-world problems. In this article, we will address a practical scenario involving the expansion of a parking lot. The core of the problem revolves around determining the amount by which the length and width of a rectangular parking lot should be increased to achieve a specific increase in its area. This involves algebraic concepts and problem-solving strategies that are commonly encountered in mathematics. Let's embark on this exploration to understand the problem, the methodology, and the step-by-step solution.
Understanding the Parking Lot Expansion Problem
In this mathematical challenge, parking lot expansion is the central theme. The problem describes a scenario where the owner of an office building intends to enlarge the parking lot. Currently, the parking lot has a rectangular shape with dimensions of 120 feet in length and 80 feet in width. The owner plans to expand the parking lot by adding the same amount to both the length and the width. This expansion aims to increase the area of the parking lot by a specific amount, which is 4,400 square feet. The main question we need to answer is: By how many feet should the length and width of the parking lot be expanded to achieve the desired area increase?
To solve this problem effectively, we need to translate the given information into mathematical terms. We'll use variables to represent the unknown quantities and set up an equation that describes the relationship between the original dimensions, the expansion, and the increase in area. This algebraic approach will allow us to find the numerical solution and determine the exact amount by which the parking lot's length and width should be expanded.
Setting Up the Equation for Parking Lot Expansion
To solve the parking lot expansion problem, we need to establish a mathematical equation that accurately represents the given information. Let's define the variable 'x' as the amount (in feet) by which both the length and width of the parking lot will be increased. The original dimensions of the parking lot are 120 feet in length and 80 feet in width. After the expansion, the new length will be (120 + x) feet, and the new width will be (80 + x) feet. The original area of the parking lot can be calculated by multiplying its original length and width, which is 120 feet * 80 feet = 9,600 square feet. The expanded area will be (120 + x) feet * (80 + x) feet.
The problem states that the expansion will increase the area of the parking lot by 4,400 square feet. This means that the expanded area is equal to the original area plus the increase in area. We can express this relationship as an equation:
(120 + x)(80 + x) = 9,600 + 4,400
This equation forms the foundation for solving the problem. It represents the relationship between the original dimensions, the expansion amount 'x,' and the desired increase in area. In the next step, we will simplify and solve this equation to find the value of 'x,' which will tell us by how many feet the length and width of the parking lot should be expanded.
Solving the Quadratic Equation for Parking Lot Expansion
Now that we have established the equation (120 + x)(80 + x) = 9,600 + 4,400, the next step is to solve it for 'x.' This equation is a quadratic equation, which means it involves a variable raised to the power of 2. Solving quadratic equations typically involves several algebraic steps.
First, let's simplify the equation by expanding the left side:
(120 + x)(80 + x) = 120 * 80 + 120 * x + 80 * x + x^2
= 9,600 + 200x + x^2
Now, let's simplify the right side of the equation:
9,600 + 4,400 = 14,000
So, the equation becomes:
9,600 + 200x + x^2 = 14,000
To solve a quadratic equation, we need to set it to zero. Subtract 14,000 from both sides of the equation:
x^2 + 200x + 9,600 - 14,000 = 0
x^2 + 200x - 4,400 = 0
Now, we have a standard quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 200, and c = -4,400. We can solve this equation using the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
Plugging in the values, we get:
x = [-200 ± sqrt(200^2 - 4 * 1 * (-4,400))] / (2 * 1)
x = [-200 ± sqrt(40,000 + 17,600)] / 2
x = [-200 ± sqrt(57,600)] / 2
x = [-200 ± 240] / 2
This gives us two possible solutions for x:
x1 = (-200 + 240) / 2 = 40 / 2 = 20
x2 = (-200 - 240) / 2 = -440 / 2 = -220
Since the expansion amount cannot be negative, we discard the negative solution (-220). Therefore, the solution for x is 20.
Determining the Expansion Amount for the Parking Lot
After solving the quadratic equation, we have found that x = 20. In the context of the parking lot expansion problem, 'x' represents the amount (in feet) by which both the length and width of the parking lot should be increased. Therefore, the length and width of the parking lot should be expanded by 20 feet each.
To verify this solution, we can calculate the new dimensions of the parking lot after the expansion:
New Length = Original Length + x = 120 feet + 20 feet = 140 feet
New Width = Original Width + x = 80 feet + 20 feet = 100 feet
Now, let's calculate the new area of the parking lot:
New Area = New Length * New Width = 140 feet * 100 feet = 14,000 square feet
To confirm that the area increase matches the problem's requirement, we subtract the original area from the new area:
Area Increase = New Area - Original Area = 14,000 square feet - 9,600 square feet = 4,400 square feet
The area increase of 4,400 square feet matches the problem statement, which confirms that our solution is correct. Therefore, to achieve the desired increase in area, the owner of the office building should expand both the length and width of the parking lot by 20 feet.
In conclusion, we have successfully solved the parking lot expansion problem by setting up and solving a quadratic equation. This problem exemplifies how mathematical principles can be applied to real-world scenarios involving geometric expansions.
Real-World Applications and Implications of Parking Lot Expansion
The parking lot expansion problem we solved is not just a theoretical exercise; it has real-world applications and implications in various fields. Understanding how to calculate the expansion of areas is crucial in construction, urban planning, and property management. Let's explore some of these applications and the broader implications of such calculations.
In construction and urban planning, developers and city planners often need to optimize the use of space. This may involve expanding existing parking facilities or designing new ones. Accurate calculations are essential to ensure that the expanded or new parking lot meets the required capacity while adhering to zoning regulations and land-use restrictions. Overestimating the expansion can lead to wasted space and resources, while underestimating it can result in insufficient parking spaces.
Property management is another area where such calculations are valuable. Property owners and managers may need to expand parking facilities to accommodate the growing needs of tenants or customers. Calculating the expansion accurately helps them make informed decisions about the scope of the project and the associated costs. It also ensures that the expansion aligns with the property's overall development plan and budget.
Moreover, the problem-solving approach used in this scenario can be applied to a wide range of similar situations. For example, it can be used to calculate the expansion of gardens, fields, or any rectangular area. The underlying mathematical principles remain the same, making this knowledge versatile and applicable across various contexts.
In addition to the practical applications, this problem highlights the importance of mathematical literacy in everyday life. Being able to translate real-world scenarios into mathematical equations and solve them empowers individuals to make informed decisions and solve problems effectively. It also fosters critical thinking and analytical skills, which are valuable in various aspects of life.
Conclusion: The Power of Mathematics in Practical Problems
In conclusion, the parking lot expansion problem serves as a compelling example of how mathematical principles can be applied to solve practical, real-world challenges. By translating the problem into an algebraic equation and employing problem-solving strategies, we successfully determined the amount by which the length and width of the parking lot should be expanded to achieve a specific increase in area. This exercise underscores the significance of mathematics in addressing everyday scenarios and making informed decisions.
Throughout our exploration, we've seen how setting up an equation, simplifying it, and solving for the unknown variable is a powerful approach in mathematics. We used the quadratic formula, a fundamental tool for solving quadratic equations, and interpreted the results in the context of the problem. The solution we arrived at not only answers the specific question but also provides insights into the broader applications of such calculations in fields like construction, urban planning, and property management.
Furthermore, this problem illustrates the value of mathematical literacy in fostering critical thinking and analytical skills. Being able to approach problems logically, break them down into manageable parts, and apply mathematical concepts is a valuable skill in various aspects of life. It empowers individuals to make sound judgments and tackle challenges effectively.
In summary, the parking lot expansion problem showcases the power of mathematics in bridging the gap between abstract concepts and concrete applications. It serves as a reminder that mathematics is not just a theoretical discipline but a practical tool that can help us navigate and understand the world around us. By embracing mathematical thinking, we can become more effective problem solvers and decision-makers in our personal and professional lives.
