Select ALL The Correct Answers.Martha Works At A Small Jewelry Store. She Designs 2 Rings In The First Hour. Every Additional Hour, She Designs 3 New Rings. Select All The Functions That Can Be Used To Find The Number Of Rings, R ( N R(n R ( N ], She

Introduction

Martha is a skilled jewelry designer who works at a small store. She has a unique way of designing rings, and her productivity increases with each passing hour. In this article, we will explore the functions that can be used to find the number of rings, denoted as $r(n)$, that Martha designs in a given number of hours, $n$. We will analyze her design process and identify the mathematical functions that can be used to model her productivity.

Martha's Design Process

Martha designs 2 rings in the first hour. Every additional hour, she designs 3 new rings. This means that her design process can be described as follows:

• In the first hour, Martha designs 2 rings.
• In the second hour, Martha designs 2 (from the first hour) + 3 (new rings) = 5 rings.
• In the third hour, Martha designs 5 (from the previous hours) + 3 (new rings) = 8 rings.
• In the fourth hour, Martha designs 8 (from the previous hours) + 3 (new rings) = 11 rings.

As we can see, Martha's design process follows a pattern where she adds 3 new rings to the total number of rings she has designed in the previous hours.

Mathematical Functions

There are several mathematical functions that can be used to model Martha's design process. Let's explore some of them:

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In Martha's case, the difference between any two consecutive terms is 3 (the number of new rings she designs in each hour). Therefore, we can model her design process using an arithmetic sequence:

$$r(n) = 2 + 3(n-1)$$

where $r(n)$ is the number of rings Martha designs in $n$ hours.

Linear Function

A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In Martha's case, the slope is 3 (the number of new rings she designs in each hour), and the y-intercept is 2 (the number of rings she designs in the first hour). Therefore, we can model her design process using a linear function:

$$r(n) = 3n + 2$$

Quadratic Function

A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In Martha's case, we can model her design process using a quadratic function:

$$r(n) = n^2 + 2n + 2$$

Exponential Function

An exponential function is a function that can be written in the form $f(x) = ab^x$, where $a$ and $b$ are constants. In Martha's case, we can model her design process using an exponential function:

$$r(n) = 2(1.5)^n$$

Conclusion

In conclusion, Martha's design process can be modeled using several mathematical functions, including arithmetic sequences, linear functions, quadratic functions, and exponential functions. Each of these functions provides a different perspective on her design process and can be used to predict the number of rings she will design in a given number of hours.

Key Takeaways

• Martha designs 2 rings in the first hour and 3 new rings in each additional hour.
• The number of rings Martha designs in $n$ hours can be modeled using arithmetic sequences, linear functions, quadratic functions, and exponential functions.
• Each of these functions provides a different perspective on Martha's design process and can be used to predict the number of rings she will design in a given number of hours.

Practice Problems

1. Find the number of rings Martha designs in 5 hours using the arithmetic sequence model.
2. Find the number of rings Martha designs in 3 hours using the linear function model.
3. Find the number of rings Martha designs in 4 hours using the quadratic function model.
4. Find the number of rings Martha designs in 2 hours using the exponential function model.

Answers

1. $r(5) = 2 + 3(5-1) = 16$
2. $r(3) = 3(3) + 2 = 11$
3. $r(4) = 4^2 + 2(4) + 2 = 26$
4. $r(2) = 2(1.5)^2 = 4.5$
   Q&A: Modeling Martha's Ring Designs =====================================

Introduction

In our previous article, we explored the mathematical functions that can be used to model Martha's ring design process. We discussed arithmetic sequences, linear functions, quadratic functions, and exponential functions, and how they can be used to predict the number of rings she will design in a given number of hours. In this article, we will answer some frequently asked questions about Martha's design process and provide additional insights into her productivity.

Q: What is the difference between an arithmetic sequence and a linear function?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In Martha's case, the difference between any two consecutive terms is 3 (the number of new rings she designs in each hour). A linear function, on the other hand, is a function that can be written in the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In Martha's case, the slope is 3 (the number of new rings she designs in each hour), and the y-intercept is 2 (the number of rings she designs in the first hour).

Q: How can I use the quadratic function to find the number of rings Martha designs in 4 hours?

A: To use the quadratic function to find the number of rings Martha designs in 4 hours, you can plug in $n = 4$ into the equation $r(n) = n^2 + 2n + 2$. This gives you:

$$r(4) = 4^2 + 2(4) + 2 = 26$$

Q: What is the advantage of using the exponential function to model Martha's design process?

A: The exponential function provides a more accurate model of Martha's design process because it takes into account the fact that her productivity increases exponentially with each passing hour. This means that the number of rings she designs in each hour increases by a fixed percentage, rather than a fixed number.

Q: Can I use the arithmetic sequence model to find the number of rings Martha designs in 6 hours?

A: Yes, you can use the arithmetic sequence model to find the number of rings Martha designs in 6 hours. To do this, you can plug in $n = 6$ into the equation $r(n) = 2 + 3(n-1)$. This gives you:

$$r(6) = 2 + 3(6-1) = 19$$

Q: How can I use the linear function to find the number of rings Martha designs in 5 hours?

A: To use the linear function to find the number of rings Martha designs in 5 hours, you can plug in $n = 5$ into the equation $r(n) = 3n + 2$. This gives you:

$$r(5) = 3(5) + 2 = 17$$

Q: What is the difference between the arithmetic sequence model and the quadratic function model?

A: The arithmetic sequence model assumes that Martha's productivity increases by a fixed number (3 new rings) in each hour, while the quadratic function model assumes that her productivity increases by a fixed percentage in each hour. The quadratic function model provides a more accurate model of Martha's design process because it takes into account the fact that her productivity increases exponentially with each passing hour.

Conclusion

In conclusion, Martha's design process can be modeled using several mathematical functions, including arithmetic sequences, linear functions, quadratic functions, and exponential functions. Each of these functions provides a different perspective on her design process and can be used to predict the number of rings she will design in a given number of hours. By understanding the strengths and weaknesses of each model, you can choose the one that best fits your needs and make more accurate predictions about Martha's productivity.

Key Takeaways

• Martha designs 2 rings in the first hour and 3 new rings in each additional hour.
• The number of rings Martha designs in $n$ hours can be modeled using arithmetic sequences, linear functions, quadratic functions, and exponential functions.
• Each of these functions provides a different perspective on Martha's design process and can be used to predict the number of rings she will design in a given number of hours.

Practice Problems

1. Find the number of rings Martha designs in 7 hours using the arithmetic sequence model.
2. Find the number of rings Martha designs in 3 hours using the linear function model.
3. Find the number of rings Martha designs in 5 hours using the quadratic function model.
4. Find the number of rings Martha designs in 2 hours using the exponential function model.

Answers

1. $r(7) = 2 + 3(7-1) = 22$
2. $r(3) = 3(3) + 2 = 11$
3. $r(5) = 5^2 + 2(5) + 2 = 37$
4. $r(2) = 2(1.5)^2 = 4.5$