5.12.1 Modeling Exponential Decay

Algebra 15.12.1 Modeling Exponential Decay

Search for key terms or text.

Warm Up

Jada received a gift of $180. In the first week, she spent a third of the gift money. She continues spending a third of what is left each week thereafter.

Which equation best represents the amount of gift money $g$ , in dollars, she has after $t$ weeks?

a. $g = 180 - \frac{1}{3} t$

b. $g = 180 \cdot (\frac{1}{3})^{t}$

c. $g = \frac{1}{3} \cdot 180^{t}$

d. $g = 180 \cdot (\frac{2}{3})^{t}$

Compare your answer:

$g = 180 \cdot (\frac{2}{3})^{t}$

Explain your reasoning.

Compare your answer:

Jada's initial gift is $180. Then, she spends a third $\frac{1}{3}$ of the whole amount each week. This means that she has $\frac{2}{3}$ left after each week because $1 - \frac{1}{3} = \frac{2}{3}$ (the whole amount minus a third).
When writing the equation, $a = 180$ because that is the starting amount she was given. Then, because she spends $\frac{1}{3}$ each week, that means only $\frac{2}{3}$ is leftover. so, $b = \frac{2}{3}$ . The equation that results is $g = 180 \cdot (23)^{t}$ .

Previous Next

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information

If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/algebra-1/pages/about-this-course
If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

Use the information below to generate a citation. We recommend using a citation tool such as this one.
- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
- Publisher/website: OpenStax
- Book title: Algebra 1
- Publication date: Jun 4, 2025
- Location: Houston, Texas
- Book URL: https://openstax.org/books/algebra-1/pages/about-this-course
- Section URL: https://openstax.org/books/algebra-1/pages/5-12-1-modeling-exponential-decay

© May 21, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.