Clarence's Magazine Subscriptions: Calculating Earnings

Jan 16, 2026 by Editorial Team 56 views

Hey guys! Let's dive into a cool math problem about Clarence and his magazine subscriptions. This is a great real-world example of how functions work. We'll break down the scenario step-by-step, making sure it's super clear and easy to understand. So, grab your coffee, and let's get started!

Understanding the Scenario: Clarence and His Magazine Sales

Alright, so here's the deal: Clarence is hustling, selling yearly subscriptions to a popular magazine. He's got a pretty consistent gig, selling between 10 and 25 subscriptions every week. That's a solid range, and it gives us some definite numbers to work with. Now, the awesome thing is, that we have a function to figure out how much money Clarence is raking in. The function is f(t) = 48t, and it's the key to unlocking the puzzle. Now, let's explore all the components. First, there is the number of subscriptions, represented as t. Secondly, f(t) is a function and it represents the amount of money earned from t number of subscriptions. Lastly, Clarence will get paid for each subscription at a rate of $48. Get ready because we are going to calculate Clarence's earnings.

To make this super clear, t here represents the number of subscriptions Clarence sells each week. So, if Clarence sells 15 subscriptions in a week, t would be 15. The function f(t), in turn, tells us how much money he makes based on the number of subscriptions he sells. In this case, each subscription is worth $48. So, the function multiplies the number of subscriptions (t) by $48 to give us Clarence's total earnings for that week. Let's say Clarence sells the bare minimum, which is 10 subscriptions. The amount he earns that week will be f(10) = 48 * 10 = $480. If he is on a roll and sells 25 subscriptions, the amount he earns will be f(25) = 48 * 25 = $1200. This is pretty sweet, right? Clarence’s earning is totally dependent on how many subscriptions he sells. Keep in mind that his number of subscriptions can only range between 10 and 25. He will never earn more or less than that range. This means the range of the function is all the values that the function can output.

Let’s summarize the scenario. Clarence is a subscription salesman, and he gets $48 for each subscription he sells. He sells at least 10 and no more than 25 subscriptions each week. This implies that there are constraints to the number of subscriptions Clarence can sell, and this will consequently affect his weekly earnings. The function f(t) = 48t helps us to calculate Clarence’s earnings. The goal is to determine the range of the function. Now we can proceed.

Breakdown of the Function

The function f(t) = 48t is a linear function. This is because the variable t (representing the number of subscriptions) is raised to the power of 1. It is a simple function, where the number of subscriptions is multiplied by $48 to find out his earnings. The main feature of this function is that it shows a direct relationship between the number of subscriptions and the amount earned. The more subscriptions Clarence sells, the more money he makes. It is important to know that functions like these help us model real-world situations, allowing us to make predictions or understand the relationships between different variables. So, f(t) helps us calculate the earning from t subscriptions.

Calculating the Range: Finding Clarence's Earnings

Alright, time to crack the main question: What is the range of the function f(t)? Remember, the range is the set of all possible output values of the function. In our case, it's all the possible amounts of money Clarence can earn in a week. We know he sells a minimum of 10 subscriptions and a maximum of 25. Therefore, we can find the range by calculating the earnings at both extremes. This will give us the lower and upper limits of his weekly income. Finding the lower and upper limits is the key. To find the minimum earnings, we'll plug in the minimum number of subscriptions, which is 10, into the function: f(10) = 48 * 10 = $480. So, the lowest amount Clarence can earn in a week is $480. Now, let's find the maximum earnings by plugging in the maximum number of subscriptions, which is 25: f(25) = 48 * 25 = $1200. This means the highest amount Clarence can earn in a week is $1200. Therefore, the range of the function f(t) is all the values between $480 and $1200, inclusive. We can express this as $480 ≤ f(t) ≤ $1200. This means Clarence will never earn less than $480 or more than $1200 in a week. The range is the possible output of a function, and we can only figure it out if we know the domain. The domain is the possible input of a function, and in this case, the domain is the number of subscriptions that Clarence can sell.

Step-by-Step Calculation

Let's go through the calculation step by step to make sure everything's crystal clear:

Identify the Minimum Subscriptions: Clarence sells a minimum of 10 subscriptions.
Calculate Minimum Earnings: f(10) = 48 * 10 = $480
Identify the Maximum Subscriptions: Clarence sells a maximum of 25 subscriptions.
Calculate Maximum Earnings: f(25) = 48 * 25 = $1200
Determine the Range: The range is from $480 to $1200, inclusive ($480 ≤ f(t) ≤ $1200).

Pretty straightforward, right? This means that every week, Clarence will always earn an amount within this range. The range is not just the minimum and maximum earnings but also all the earnings in between. Since the function is linear, the earnings increase steadily as the number of subscriptions increases. So, if he sells 11 subscriptions, he makes $528; if he sells 12 subscriptions, he makes $576, and so on.

Conclusion: Clarence's Weekly Earning Potential

So, there you have it, guys! We've successfully calculated the range of Clarence's earnings. He's got a pretty good gig, making anywhere from $480 to $1200 a week, depending on how many subscriptions he sells. It's a great example of how simple mathematical functions can model real-world scenarios and help us understand relationships between different values. Keep in mind that the range is not just the minimum and maximum earnings but also all the earnings in between. Since the function is linear, the earnings increase steadily as the number of subscriptions increases. We also went through a step-by-step guide to determine the range. Always start by identifying the minimum and maximum inputs, and then, you will be able to determine the range. Understanding the range helps us to predict and understand the possible outcomes.

This also shows how important it is to understand the boundaries of a function. The domain (the number of subscriptions) has a direct impact on the range (the earnings). Change the domain, and you'll change the range. This connection is super important in mathematics and in many real-world applications. By knowing the constraints (in this case, the minimum and maximum number of subscriptions), we can determine the possible outputs (Clarence's earnings). Understanding the constraints is the key to mastering the range!

Final Thoughts

I hope you enjoyed this journey into Clarence's world of magazine subscriptions! It is important to know that functions are not just about numbers and equations; they're about understanding how things work in the real world. Keep practicing, and you'll become a function whiz in no time. If you have any questions, don't hesitate to ask! Happy calculating!

This simple math problem illustrates a crucial concept: that the domain of a function directly influences its range. The domain, representing the number of subscriptions Clarence sells (between 10 and 25), dictates the possible earnings (the range). Changing the domain would inherently shift the range. Understanding this relationship is fundamental in mathematics and critical for analyzing numerous real-life applications. The limits of the input (domain) translate to limits on the output (range). So, when you are analyzing a math problem, be mindful of the constraints. This will allow you to determine the possible outputs accurately. Math is cool, and it is even cooler when applied to real-life situations like Clarence's subscription business. Understanding concepts like range and domain is very important to solve these problems.