Parking Lot Expansion Problem Solving With Quadratic Equations
In the realm of real estate and property management, optimizing space is a crucial endeavor. This article delves into a practical mathematical problem encountered when expanding a parking lot. We will explore how to determine the precise amount by which the length and width of a parking lot should be increased to achieve a desired expansion in area. This problem combines geometric principles with algebraic problem-solving, offering a valuable case study in applied mathematics.
Understanding the Parking Lot Expansion Problem
Let's consider a scenario where the owner of an office building intends to enlarge the parking lot to accommodate more vehicles. The existing parking lot has dimensions of 120 feet in length and 80 feet in width. The expansion plan aims to increase the area of the parking lot by 4,400 square feet. The challenge is to find out by how many feet the length and width should be extended, assuming the same amount is added to both dimensions. This problem allows us to apply mathematical concepts such as area calculation, quadratic equations, and problem-solving strategies in a real-world context.
Setting Up the Equation: A Step-by-Step Approach
To solve this problem effectively, we need to translate the word problem into a mathematical equation. The first step is to represent the unknown quantity with a variable. Let 'x' represent the amount (in feet) by which both the length and the width of the parking lot will be increased. The new dimensions of the parking lot will then be (120 + x) feet in length and (80 + x) feet in width. The area of the expanded parking lot can be calculated by multiplying the new length and width: (120 + x)(80 + x).
The original area of the parking lot is simply the product of its original dimensions, which is 120 feet multiplied by 80 feet, resulting in 9,600 square feet. The expansion is designed to increase the area by 4,400 square feet. Therefore, the area of the expanded parking lot will be the original area plus the increase, which is 9,600 + 4,400 = 14,000 square feet. Now, we can set up the equation by equating the expression for the expanded area to the total area after expansion:
(120 + x)(80 + x) = 14,000
This equation is a quadratic equation, which we will need to solve to find the value of 'x'. Solving quadratic equations often involves expanding the equation, rearranging it into standard form, and then using methods such as factoring, completing the square, or applying the quadratic formula. In the next section, we will delve into the algebraic steps to solve this equation and determine the value of 'x'.
Solving the Quadratic Equation: Finding the Expansion Amount
Now that we have set up the equation (120 + x)(80 + x) = 14,000, the next step is to solve for 'x'. This involves algebraic manipulation to bring the equation into a standard quadratic form (ax^2 + bx + c = 0). First, we expand the left side of the equation:
120 * 80 + 120 * x + 80 * x + x^2 = 14,000
This simplifies to:
9600 + 200x + x^2 = 14,000
Next, we rearrange the equation to set it to zero, which is the standard form for a quadratic equation. We subtract 14,000 from both sides:
x^2 + 200x + 9600 - 14,000 = 0
This further simplifies to:
x^2 + 200x - 4400 = 0
Now we have a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 200, and c = -4400. To solve this equation, we can use the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
Substituting the values, we get:
x = [-200 ± sqrt(200^2 - 4 * 1 * -4400)] / (2 * 1)
x = [-200 ± sqrt(40000 + 17600)] / 2
x = [-200 ± sqrt(57600)] / 2
x = [-200 ± 240] / 2
This gives us two possible solutions for x:
x_1 = (-200 + 240) / 2 = 40 / 2 = 20
x_2 = (-200 - 240) / 2 = -440 / 2 = -220
Since the expansion amount cannot be negative, we discard the negative solution. Therefore, the value of x is 20 feet. This means that both the length and width of the parking lot should be increased by 20 feet to achieve the desired expansion in area.
Verifying the Solution: Ensuring Accuracy and Practicality
After obtaining a solution to a mathematical problem, it's crucial to verify its accuracy and practicality, especially in real-world applications. In this parking lot expansion scenario, we found that increasing both the length and width by 20 feet would result in the desired 4,400 square feet increase in area. To verify this solution, we can calculate the new dimensions and the new area of the parking lot.
The original dimensions were 120 feet in length and 80 feet in width. Adding 20 feet to each dimension gives us a new length of 120 + 20 = 140 feet and a new width of 80 + 20 = 100 feet. The new area is then 140 feet multiplied by 100 feet, which equals 14,000 square feet.
The original area was 120 feet * 80 feet = 9,600 square feet. The increase in area is the new area minus the original area, which is 14,000 - 9,600 = 4,400 square feet. This matches the desired expansion in area, confirming the accuracy of our solution.
Practical Considerations in Real-World Application
Beyond the mathematical accuracy, it is important to consider the practical implications of the solution. In a real-world parking lot expansion project, several factors come into play. For example, the layout of the parking lot, the existing infrastructure, and local zoning regulations can influence the feasibility of the expansion. The shape and dimensions of the expanded area must also be practical for parking vehicles and maneuvering within the lot.
Furthermore, the cost of the expansion, including materials, labor, and any necessary permits, needs to be considered. The owner would also need to think about the impact on traffic flow and pedestrian access. A well-planned expansion should not only increase the parking capacity but also maintain or improve the overall functionality and safety of the parking lot.
In addition to the physical aspects, the owner might also consider future needs and potential further expansions. A long-term perspective can help in making decisions that are both cost-effective and sustainable. Therefore, while the mathematical solution provides a precise answer to the problem, real-world applications require a holistic approach that considers various practical and logistical factors.
Conclusion: The Intersection of Mathematics and Real-World Problem Solving
In conclusion, the problem of expanding a parking lot's dimensions illustrates how mathematical principles are applied in real-world scenarios. By translating the problem into a quadratic equation, we were able to determine the precise amount by which the length and width should be increased to achieve the desired expansion in area. The solution, 20 feet, was derived using the quadratic formula and verified through area calculations.
However, the mathematical solution is just one aspect of the overall problem. Real-world applications require a more comprehensive approach that takes into account practical considerations such as the physical layout, cost, zoning regulations, and future needs. The intersection of mathematics and practical considerations highlights the importance of problem-solving skills that go beyond numerical calculations. It involves critical thinking, logical reasoning, and the ability to integrate various factors to arrive at an optimal solution.
This parking lot expansion problem serves as a valuable case study in applied mathematics. It demonstrates how algebraic concepts, such as quadratic equations, can be used to solve practical problems in property management and real estate. It also underscores the significance of verifying solutions and considering real-world constraints to ensure the feasibility and effectiveness of any proposed expansion. By combining mathematical rigor with practical insights, we can approach such problems with confidence and achieve successful outcomes.
