Determine The Linear Function For Basketball Ticket Costs

In the realm of mathematics, particularly in linear functions, real-world applications provide a practical understanding of abstract concepts. One such application is determining the cost of tickets for a basketball game, where a fixed price per ticket is combined with a service fee. This scenario perfectly illustrates how a linear function can model the total cost based on the number of tickets purchased. This article aims to dissect this problem, providing a comprehensive explanation of how to derive the linear function that represents the total cost of ordering basketball tickets online.

When approaching a problem involving linear functions, it's crucial to identify the key components: the variable cost (the cost that changes with the number of items), the fixed cost (a constant cost regardless of the number of items), and the relationship between them. In the context of basketball tickets, the price per ticket is the variable cost, while the service fee is the fixed cost. The total cost is then the sum of these two components. Understanding this breakdown is the first step in formulating the linear function. We'll delve into how to translate the given information—the cost of 5 tickets being $108.00 including a $5.50 service fee—into a mathematical equation. This involves setting up an equation, solving for the unknown price per ticket, and then constructing the complete linear function. The process will highlight the importance of accurately interpreting the problem's context and applying algebraic principles to arrive at the correct solution. Furthermore, we'll explore how this linear function can be used to predict the cost for any number of tickets, making it a valuable tool for both the ticket vendor and the customer.

To tackle this mathematical problem effectively, let's break it down into smaller, more manageable parts. We're given that tickets to a basketball game are available online at a set price per ticket, and there's an additional $5.50 service fee applied to the entire order. We also know that the total cost for ordering 5 tickets comes out to be $108.00. Our ultimate goal is to determine the linear function that represents c, the total cost, when x represents the number of tickets ordered. This linear function will allow us to calculate the total cost for any number of tickets, providing a clear and concise model for ticket pricing.

The first step in this breakdown is recognizing the components of the total cost. The total cost c is comprised of two main parts: the cost of the tickets themselves and the service fee. The cost of the tickets is directly proportional to the number of tickets purchased, x. This means that if we double the number of tickets, the cost of the tickets also doubles. This proportional relationship is a key characteristic of linear functions. The service fee, on the other hand, is a fixed cost. It remains constant regardless of the number of tickets ordered. This fixed cost is what shifts the linear function away from the origin (0,0) on a graph.

With this understanding, we can start to formulate the equation. Let's represent the price per ticket as p. The total cost of the tickets alone would then be p * x, where x is the number of tickets. Adding the service fee of $5.50 gives us the total cost c. So, we can write the equation as c = p * x + 5.50. This equation is the basic form of a linear function, where p is the slope (the rate of change of cost with respect to the number of tickets) and 5.50 is the y-intercept (the cost when no tickets are purchased). The next step involves using the given information—5 tickets costing $108.00—to solve for p, the price per ticket. This will give us the specific linear function for this scenario.

To find the linear function, a crucial step is determining the price per ticket. We know that ordering 5 tickets results in a total cost of $108.00, which includes the $5.50 service fee. We can use this information to calculate the price per ticket using a simple algebraic equation. Recalling the equation c = p * x + 5.50, where c is the total cost, p is the price per ticket, and x is the number of tickets, we can substitute the given values. In this case, c is $108.00 and x is 5.

Substituting these values into the equation, we get 108.00 = p * 5 + 5.50. Now, our goal is to isolate p to find the price per ticket. The first step is to subtract the service fee, $5.50, from both sides of the equation. This gives us 108.00 - 5.50 = p * 5, which simplifies to 102.50 = p * 5. This equation tells us that the cost of the 5 tickets alone, without the service fee, is $102.50. To find the price of a single ticket, we need to divide both sides of the equation by 5. This will give us 102.50 / 5 = p. Performing the division, we find that p = 20.50. This means the price per basketball ticket is $20.50.

Now that we have calculated the price per ticket, $20.50, we have a critical piece of information needed to define the linear function. This price represents the variable cost component of the total cost. It's the amount that changes directly with the number of tickets purchased. With this value, we can now complete the equation for the total cost, incorporating both the variable cost (price per ticket times the number of tickets) and the fixed cost (the service fee). This completed equation will be the linear function that represents the total cost c for any number of tickets x. The next step is to put it all together and present the final linear function.

Having determined the price per ticket to be $20.50, we can now construct the complete linear function that represents the total cost, c, when x tickets are ordered. Recall that the general form of a linear function in this context is c = p * x + 5.50, where p is the price per ticket and $5.50 is the service fee. We've already calculated p to be $20.50, so we can substitute this value into the equation. This substitution gives us the specific linear function for this basketball ticket scenario: c = 20.50x + 5.50.

This equation, c = 20.50x + 5.50, is the solution to our problem. It represents a direct and clear mathematical model for the total cost of ordering tickets online. The coefficient 20.50, which multiplies x, represents the slope of the line. In this context, the slope signifies the rate at which the total cost increases for each additional ticket purchased. For every ticket added to the order, the total cost goes up by $20.50. The constant term, 5.50, represents the y-intercept of the line. This is the point where the line crosses the y-axis on a graph, and in this scenario, it represents the fixed service fee of $5.50, which is charged regardless of the number of tickets ordered.

This linear function is incredibly useful because it allows us to easily calculate the total cost for any number of tickets. For instance, if someone wanted to order 10 tickets, they could simply substitute x = 10 into the equation: c = 20.50(10) + 5.50. This would result in c = 205.00 + 5.50, so the total cost would be $210.50. Similarly, the function could be used to determine how many tickets can be purchased for a given budget. The function provides a versatile tool for both customers and the ticket vendor, allowing for easy cost estimation and budgeting. In conclusion, understanding how to construct and interpret linear functions like this one is a valuable skill in mathematics and in everyday life, enabling us to model and predict real-world scenarios involving costs and quantities.

Now that we have established the linear function c = 20.50x + 5.50 to represent the total cost of ordering basketball tickets, we can explore its practical applications in predicting costs for various scenarios. This function isn't just an abstract mathematical concept; it's a powerful tool that allows us to quickly and accurately calculate the total cost for any number of tickets. Whether you're planning a family outing, a group event, or simply curious about the potential cost of different ticket quantities, this function provides the answer.

One of the most straightforward applications is to predict the cost for a specific number of tickets. Let's say you're planning to attend the game with a group of 8 friends. To determine the total cost, you would substitute x = 8 into the equation: c = 20.50(8) + 5.50. This simplifies to c = 164.00 + 5.50, resulting in a total cost of $169.50. This quick calculation provides you with the necessary information to budget for the event and ensure you have enough funds to cover the ticket purchase. Similarly, if you were considering inviting more friends and wanted to know the cost for 12 tickets, you would repeat the process with x = 12, leading to c = 20.50(12) + 5.50, which equals $251.50.

Another valuable application of the linear function is determining the maximum number of tickets you can purchase with a given budget. For example, suppose you have a budget of $300 and want to know how many tickets you can afford. In this case, we would set c = 300 and solve for x. The equation becomes 300 = 20.50x + 5.50. First, subtract 5.50 from both sides: 294.50 = 20.50x. Then, divide both sides by 20.50: x ≈ 14.37. Since you can't purchase a fraction of a ticket, you would round down to the nearest whole number. This means you can purchase a maximum of 14 tickets with a $300 budget. These examples illustrate the versatility of the linear function in both predicting costs and making informed purchasing decisions. It provides a clear and reliable way to understand the relationship between the number of tickets and the total cost, empowering you to plan your basketball game outing effectively.

In conclusion, this exploration of basketball ticket pricing has provided a practical demonstration of the power and utility of linear functions in real-world scenarios. By breaking down the problem into its core components—the price per ticket, the service fee, and the total cost—we were able to construct a linear function that accurately models the relationship between the number of tickets purchased and the overall expense. This linear function, c = 20.50x + 5.50, serves as a valuable tool for predicting costs, making budget decisions, and understanding the fundamental principles of linear relationships.

The process of deriving this linear function involved several key steps, each highlighting the importance of mathematical reasoning and problem-solving skills. First, we identified the variable and fixed costs, recognizing that the price per ticket was the variable component and the service fee was the fixed component. This understanding allowed us to formulate the general equation c = p * x + 5.50, where p represented the unknown price per ticket. Next, we utilized the given information—the total cost for 5 tickets—to solve for p, employing algebraic techniques to isolate the variable and determine its value. This step demonstrated the crucial role of algebraic manipulation in solving equations and extracting meaningful information. Finally, with the price per ticket calculated, we constructed the complete linear function, showcasing how to combine variable and fixed costs into a single, cohesive mathematical model.

Beyond simply deriving the function, we also explored its practical applications in predicting costs and making purchasing decisions. By substituting different values for x, the number of tickets, we were able to quickly and easily calculate the corresponding total cost, c. This demonstrated the predictive power of linear functions and their ability to provide valuable insights into real-world situations. Moreover, we showed how the function could be used to determine the maximum number of tickets that could be purchased within a given budget, further highlighting its utility in financial planning. This exercise underscores the significance of linear functions not only in mathematics but also in everyday life, enabling us to make informed decisions and navigate financial considerations with greater confidence. The ability to translate real-world scenarios into mathematical models is a fundamental skill, and this example of basketball ticket pricing serves as a clear and compelling illustration of its importance.