Finding The Maximum Number Of Passengers On An Airplane
This article delves into the mathematical problem of determining how many more passengers can board an airplane, given its seating capacity and the number of passengers already on board. We'll explore the use of inequalities to represent this scenario and solve for the unknown, providing a clear and concise explanation suitable for students and anyone interested in practical applications of mathematics. This problem illustrates a common real-world scenario where inequalities are used to model constraints and find solutions within those constraints. Understanding how to set up and solve inequalities is a valuable skill that extends beyond the classroom, finding applications in various fields such as resource management, budgeting, and capacity planning. In this specific case, we are dealing with the capacity of an airplane, but the same principles can be applied to any situation where there is a limit or maximum value.
The core concept we will be using is that of an inequality. An inequality is a mathematical statement that compares two expressions using symbols like less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). In this problem, we are dealing with a situation where the total number of passengers cannot exceed the airplane's capacity. This will lead us to use the "less than or equal to" (≤) symbol in our inequality. We will first define a variable to represent the unknown quantity – the number of additional passengers that can board the plane. Then, we will translate the given information into a mathematical inequality. The inequality will express the condition that the sum of the current passengers and the additional passengers must be less than or equal to the plane's maximum capacity. Once we have the inequality, we will use algebraic techniques to solve for the variable. This involves isolating the variable on one side of the inequality to determine the range of possible values it can take. The solution will tell us the maximum number of additional passengers that can board the plane while staying within its capacity. Finally, we will interpret the solution in the context of the problem, providing a clear answer to the original question.
Problem Statement
Li's family is preparing for a flight, and the airplane they will be flying on has a seating capacity of up to 149 passengers. Currently, there are 96 passengers already on board. The problem is to determine which inequality can be used to find out how many more people can board the plane.
Understanding the Constraints
The airplane has a limited number of seats, which means there's a maximum capacity. This constraint is crucial because it dictates the type of mathematical expression we'll use to represent the situation. In this case, the total number of passengers (current plus additional) cannot exceed 149.
Defining the Variable
To formulate the inequality, we need to define a variable. Let's use 'x' to represent the number of additional passengers who can board the plane. This variable is what we're trying to determine.
Formulating the Inequality
Now, we can translate the problem into a mathematical inequality. The current number of passengers (96) plus the additional passengers (x) must be less than or equal to the plane's capacity (149). This can be written as:
96 + x ≤ 149
This inequality accurately represents the situation. It states that the sum of the current passengers and any additional passengers must not exceed the maximum capacity of the plane.
Analyzing Other Options
The problem also presents another inequality option:
96 + x ≥ 149
This inequality is incorrect because it implies that the total number of passengers must be greater than or equal to 149, which contradicts the constraint that the plane can seat up to 149 passengers. Therefore, this option is not suitable for representing the given scenario.
Solving the Inequality
Now that we've identified the correct inequality, let's solve it to find the maximum number of additional passengers who can board the plane. The inequality is:
96 + x ≤ 149
Isolating the Variable
To solve for 'x', we need to isolate it on one side of the inequality. We can do this by subtracting 96 from both sides:
96 + x - 96 ≤ 149 - 96
This simplifies to:
x ≤ 53
Interpreting the Solution
The solution, x ≤ 53, means that the number of additional passengers ('x') must be less than or equal to 53. In practical terms, this means that a maximum of 53 more passengers can board the plane without exceeding its capacity.
Verifying the Solution
To ensure our solution is correct, we can substitute the maximum value of 'x' (which is 53) back into the original inequality:
96 + 53 ≤ 149
149 ≤ 149
This statement is true, which confirms that our solution is correct. If we were to add more than 53 passengers, the total would exceed the plane's capacity.
Real-World Implications
This problem highlights the practical application of inequalities in real-world scenarios. Airlines use these types of calculations to ensure they comply with safety regulations and capacity limits. Understanding these concepts is not only useful for solving mathematical problems but also for making informed decisions in everyday situations.
Capacity Planning
Businesses and organizations often need to plan capacity based on various constraints. For example, a restaurant needs to determine how many tables it can accommodate given the space available and fire safety regulations. A warehouse needs to calculate how much inventory it can store based on its storage capacity.
Resource Management
Inequalities can also be used in resource management. For instance, a city might need to determine how much water it can allocate to different sectors (e.g., residential, commercial, industrial) based on the total water supply and demand.
Budgeting
In personal or business budgeting, inequalities can help ensure that expenses do not exceed income. By setting up inequalities, individuals and organizations can track their spending and make adjustments as needed.
Conclusion
In this article, we successfully identified and solved the inequality that represents the problem of finding the maximum number of additional passengers who can board an airplane. We defined the variable, formulated the inequality, solved for the variable, and interpreted the solution in the context of the problem. This example illustrates the power of inequalities in modeling real-world constraints and finding solutions within those constraints. The inequality 96 + x ≤ 149 correctly models the scenario, and solving it gives us x ≤ 53, meaning a maximum of 53 more passengers can board. Understanding and applying these concepts can help solve various problems related to capacity, resources, and budgeting, making it a valuable skill in mathematics and beyond. The ability to translate real-world scenarios into mathematical expressions and solve them is a fundamental skill in problem-solving. This example demonstrates how inequalities can be used to model constraints and find solutions within those constraints. By understanding these concepts, individuals can make better decisions in various aspects of their lives, from managing personal finances to planning events.
FAQ
Why is the inequality 96 + x ≤ 149 the correct representation of the problem?
The inequality 96 + x ≤ 149 is correct because it accurately represents the constraint that the total number of passengers (current plus additional) cannot exceed the plane's capacity of 149. The "≤" symbol means "less than or equal to," which is appropriate because the total number of passengers can be 149 or less, but not more.
What does the solution x ≤ 53 mean in the context of the problem?
The solution x ≤ 53 means that the number of additional passengers ('x') can be any number less than or equal to 53. In practical terms, this means that a maximum of 53 more passengers can board the plane without exceeding its capacity. Any number of passengers less than 53 could also board, but 53 is the maximum.
How can inequalities be used in other real-world scenarios?
Inequalities are used in various real-world scenarios to model constraints and find solutions within those constraints. Some examples include:
- Capacity planning: Determining how many people or items can fit in a space given certain limitations.
- Resource management: Allocating resources (e.g., water, budget) based on demand and availability.
- Budgeting: Ensuring that expenses do not exceed income.
- Manufacturing: Optimizing production processes to meet demand while minimizing costs.
What are the key steps in solving an inequality?
The key steps in solving an inequality are:
- Identify the variable: Determine what you are trying to find and assign a variable to it.
- Formulate the inequality: Translate the problem's constraints into a mathematical inequality.
- Isolate the variable: Use algebraic operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the inequality. Remember that multiplying or dividing by a negative number reverses the inequality sign.
- Interpret the solution: Understand what the solution means in the context of the problem.
- Verify the solution: Substitute the solution back into the original inequality to ensure it is correct.
What happens if we use the incorrect inequality 96 + x ≥ 149?
If we use the inequality 96 + x ≥ 149, we would be incorrectly stating that the total number of passengers must be greater than or equal to 149. This contradicts the problem's constraint that the plane can seat up to 149 passengers. Solving this incorrect inequality would give us a solution that does not make sense in the context of the problem, as it would suggest that more passengers can board the plane than there are seats available.
