Functions to Find the Number of Rings

Oct 13, 2025 by windailys.com 47 views

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, $r(n)$ , she designs in $n$ hours.

Martha's Jewelry Design Process

Martha designs 2 rings in the first hour. Every additional hour, she designs 3 new rings. This means that her productivity increases by 3 rings per hour after the first hour. To find the number of rings she designs in $n$ hours, we need to consider her initial productivity and the additional rings she designs in each subsequent hour.

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 this case, we can use a linear function to model Martha's jewelry design process.

Let's consider the number of rings Martha designs in $n$ hours. In the first hour, she designs 2 rings, and in each subsequent hour, she designs 3 additional rings. This means that the number of rings she designs in $n$ hours is given by the linear function:

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

Simplifying this expression, we get:

r(n) = 3n - 1

This linear function represents the number of rings Martha designs in $n$ hours.

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 this case, we can use a quadratic function to model Martha's jewelry design process.

r(n) = 2 + 3n + \frac{3}{2}(n - 1)^2

Simplifying this expression, we get:

r(n) = \frac{3}{2}n^2 + \frac{1}{2}n

This quadratic function represents the number of rings Martha designs in $n$ hours.

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 this case, we can use an exponential function to model Martha's jewelry design process.

r(n) = 2 \cdot 3^{n-1}

This exponential function represents the number of rings Martha designs in $n$ hours.

Conclusion

In this article, we have explored the functions that can be used to find the number of rings, $r(n)$ , Martha designs in $n$ hours. We have considered linear, quadratic, and exponential functions, and we have shown that each of these functions can be used to model Martha's jewelry design process.

Functions to Find the Number of Rings

Linear Function: $r(n) = 3n - 1$
Quadratic Function: $r(n) = \frac{3}{2}n^2 + \frac{1}{2}n$
Exponential Function: $r(n) = 2 \cdot 3^{n-1}$

These functions can be used to find the number of rings Martha designs in $n$ hours.

Example Use Cases

Finding the Number of Rings in 5 Hours: To find the number of rings Martha designs in 5 hours, we can use any of the functions above. For example, using the linear function, we get:

r(5) = 3(5) - 1 = 14

This means that Martha designs 14 rings in 5 hours.

Finding the Number of Rings in 10 Hours: To find the number of rings Martha designs in 10 hours, we can use any of the functions above. For example, using the quadratic function, we get:

r(10) = \frac{3}{2}(10)^2 + \frac{1}{2}(10) = 155

This means that Martha designs 155 rings in 10 hours.

Conclusion

Q: What is Martha's initial productivity?

A: Martha's initial productivity is 2 rings per hour. This means that in the first hour, she designs 2 rings.

Q: How many rings does Martha design in each subsequent hour?

A: Martha designs 3 additional rings in each subsequent hour. This means that in the second hour, she designs 2 + 3 = 5 rings, in the third hour, she designs 5 + 3 = 8 rings, and so on.

Q: What is the formula for the number of rings Martha designs in n hours?

A: The formula for the number of rings Martha designs in n hours is:

Linear Function: $r(n) = 3n - 1$
Quadratic Function: $r(n) = \frac{3}{2}n^2 + \frac{1}{2}n$
Exponential Function: $r(n) = 2 \cdot 3^{n-1}$

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

A: To find the number of rings Martha designs in 5 hours using the linear function, you can plug in n = 5 into the formula:

r(5) = 3(5) - 1 = 14

This means that Martha designs 14 rings in 5 hours.

Q: How can I use the quadratic function to find the number of rings Martha designs in 10 hours?

A: To find the number of rings Martha designs in 10 hours using the quadratic function, you can plug in n = 10 into the formula:

r(10) = \frac{3}{2}(10)^2 + \frac{1}{2}(10) = 155

This means that Martha designs 155 rings in 10 hours.

Q: What is the relationship between the number of hours and the number of rings Martha designs?

A: The number of rings Martha designs is directly proportional to the number of hours she works. This means that as the number of hours increases, the number of rings she designs also increases.

Q: Can I use the exponential function to find the number of rings Martha designs in any hour?

A: Yes, you can use the exponential function to find the number of rings Martha designs in any hour. Simply plug in the value of n into the formula:

r(n) = 2 \cdot 3^{n-1}

Q: What is the advantage of using the linear function to find the number of rings Martha designs?

A: The advantage of using the linear function is that it is easy to calculate and provides a simple formula for finding the number of rings Martha designs.

Q: What is the advantage of using the quadratic function to find the number of rings Martha designs?

A: The advantage of using the quadratic function is that it provides a more accurate model of Martha's jewelry design process, especially for larger values of n.

Q: What is the advantage of using the exponential function to find the number of rings Martha designs?

A: The advantage of using the exponential function is that it provides a simple and elegant formula for finding the number of rings Martha designs, especially for larger values of n.

Conclusion

In this Q&A article, we have answered some common questions about Martha's jewelry design process and provided formulas for finding the number of rings she designs in n hours. We have also discussed the advantages and disadvantages of using different functions to model Martha's jewelry design process.