Ring Design: Functions For Martha's Jewelry Creations

Let's dive into a fun little problem about Martha, a talented jewelry designer! Martha works at a small jewelry store, and she's got a knack for creating beautiful rings. In the first hour of her workday, she designs 2 rings. As she gets into the groove, she designs 3 new rings every additional hour. Our mission is to figure out which functions can accurately represent the total number of rings, denoted as r(n), that Martha designs in n hours.

Understanding the Problem

Before we jump into the functions, let's make sure we understand the core concept. Martha's ring-designing pace isn't constant; it changes after the first hour. This means we're dealing with a situation where the initial condition is different from the subsequent pattern. She starts by creating 2 rings within the first hour of her work. This is the base upon which her subsequent production depends. Each hour after that, Martha's productivity increases. She begins to design rings more quickly, creating 3 new rings per hour. What we need to keep in mind is that after the first hour, the number of rings created is always going to be two, plus whatever amount of hours passed. Understanding the nuances of what happens after the first hour is essential. The core of the issue, therefore, is determining the best approach to expressing this pattern mathematically.

Analyzing the Given Options

Now, let's analyze the provided options to see which ones fit the bill. As we go through each function, we'll evaluate whether it correctly captures Martha's initial ring design and her subsequent increase in production. The function must describe how the total number of rings changes over time, as each new hour provides the opportunity for Martha to create more intricate and beautiful rings. In addition, the function should be able to precisely predict the quantity of rings that Martha would have designed at any given moment. We can assess whether a particular function is accurate by calculating r(n) for particular values of n and contrasting these values with the real number of rings that Martha would have designed. This iterative procedure guarantees that we choose the function that most accurately represents Martha's output.

Finding the Right Function

To properly model Martha's ring-designing process, we need a function that accounts for her initial burst of 2 rings and the subsequent consistent increase of 3 rings per hour. Let's break down how we can build such a function. We know that for the first hour (n = 1), Martha designs 2 rings. So, r(1) = 2. For every hour after that, she designs 3 more rings. This means that for n > 1, we can express the number of rings as r(n) = 2 + 3(n - 1). Why (n - 1)? Because the initial 2 rings cover the first hour, and we only need to account for the additional hours multiplied by the rate of 3 rings per hour. Simplifying this expression, we get r(n) = 2 + 3n - 3, which further simplifies to r(n) = 3n - 1. This linear function encapsulates Martha's work pattern, establishing a link between the number of hours she puts in and the total quantity of rings that she designs. It allows us to predict Martha's creative output over any period of time with precision. Now, in order to accurately capture Martha's ring-designing process, it is essential to test this function and make sure it works for different values of 'n.' For example, we know that if Martha designs rings for two hours, she makes 2 rings in the first hour and an additional 3 rings in the second. We will therefore have a total of 5 rings. To determine whether the function r(n) = 3n - 1 accurately reflects this information, we may replace n with 2. This yields r(2) = 3(2) - 1 = 6 - 1 = 5, which is consistent with the information that we have. This verification guarantees that the function accurately describes Martha's production following the first hour. In order to further validate the model, further testing may be carried out using additional values of n. In order to ensure that the function accurately depicts Martha's rate of ring design throughout her job, each experiment adds to the degree of confidence. This thorough method confirms that the linear function r(n) = 3n - 1 is a strong contender for accurately representing Martha's artistic output.

Why Other Options Might Not Work

It's crucial to understand why other options might not be suitable for representing Martha's ring-designing process. Let's consider a hypothetical function that simply states r(n) = 3n. This function would imply that Martha designs 3 rings every hour, starting from the very first hour. However, we know that Martha designs only 2 rings in the first hour, making this function inaccurate. Similarly, a function like r(n) = 2n would suggest that Martha consistently designs 2 rings per hour, failing to account for the increased rate of 3 rings per hour after the initial hour. The key is to recognize that Martha's ring-designing pace is not uniform throughout her workday. The first hour has a different output compared to the subsequent hours. Any function that doesn't accommodate this variation cannot accurately represent Martha's actual ring-designing process. Therefore, when evaluating potential functions, it's essential to carefully consider how they handle the initial condition and the subsequent pattern of Martha's work.

Conclusion

In summary, the function that accurately describes the number of rings Martha designs in n hours must consider her initial output of 2 rings in the first hour and the subsequent increase of 3 rings per hour. By carefully analyzing the problem and building a function that accounts for these factors, we can accurately model Martha's ring-designing process. Remember, the key is to break down the problem into smaller parts, understand the underlying patterns, and then translate those patterns into a mathematical expression.

For further learning on mathematical functions and modeling, you can visit Khan Academy's Algebra I section.