INTRODUCTION
Life is full of hazards and complications. Algebraic functions help define the problems and provide a concise visual aid. Exponential functions are so good at explaining real-world issues that they are utilized in every industry. From healthcare to retail sales, they can drive decision-making. For example, the healthcare industry uses radiation to treat several health conditions. However, working with radioactive material is extremely dangerous. Therefore, identifying the radioactive half-life of the material is essential for safe handling. A radioactive half-life describes when the material will be safe to handle without protection.
Below is an analysis of the half-life for a fictitious radioactive material. Table 1 summarizes data collected from the radioactive samples to understand the rate of decay. Figure 1 is a scatter plot with an exponential regression line of best fit to calculate and predict decay rate (McKenzie-McHarg, 2020). The half-life is ≈ 5.3 months. After ≈ 12.3 months, the weight of the radioactive material will be ≈ 40 grams. During month 30, the weight of the material will be ≈ 4 grams.
CALCULATING RADIOACTIVE MATERIAL DECAY USING AN EXPONENTIAL FUNCTION
The expression used for the exponential line of best fit is 𝑓(𝑥) = 𝑎(𝑏)^𝑥 (McKenzieMcHarg, 2020). Let 𝑎 represent the weight in grams, let 𝑏 represent the exponential base or growth rate, and 𝑥 represents the time variable. When calculating the half-life, 𝑏 is less than one and greater than zero, showing an exponential decay (Edwards, 2011). Figure 1 shows the half-life is ≈ 5.3 months by looking at the x coordinate for the y value of 100. The y-intercept is 200.35 and shows where x meets the y axis, indicating that at month zero, the weight in grams of the radioactive material is 200.35 and from there declines in weight. The domain is all real numbers. The range is (0,∞). The horizontal asymptote is y = 0, meaning as the values of x decrease, the curve arbitrarily gets closer to 0. The function is neither odd nor even. However, it is a one-to-one function that has an inverse.
DATA VISUALIZATION OF RADIOACTIVE MATERIAL DECAY
Table 1
Radioactive Material Decay Data
Time - Months | Weight - Grams |
0 | 200 |
1 | 176.6 |
2 | 151.8 |
3 | 138.8 |
4 | 118.2 |
5 | 103.6 |
6 | 91 |
Note. Data provided by Colorado State University - Global Campus.
Figure 1
Working with Radioactive Material - Scatter Plot with Regression Line
CONCLUSION
This experiment shows that the half-life of the radioactive material studied is ≈ 5.3 months. In addition, at month 12.3, Figure 1 shows the weight will be ≈ 40 grams. After 30 months, Figure 1 shows that the remaining weight of the radioactive material will be ≈ 4 grams. Therefore, take safety precautions to avoid radiation exposure as the material decays when handling the material.
Radioactive decay is not the only problem that can apply algebraic functions for solving. For example, banks use them to calculate compound interest, governments use them to solve population growth, programmers use them to express commands and calculate algorithmic efficiency, and supply chains use them to solve supply and demand. These are a few examples of practical applications for algebra functions. There are many other creative ways to use them. Contact me today to create a data story designed for you or your organization.
About the Author
Sondra Hoffman is a seasoned MIS professional with over ten years of experience in strategic planning, implementation, and optimization of MIS solutions. She is passionate about helping small businesses thrive through technology and data management. Connect with her on LinkedIn to learn more about her professional background.
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REFERENCES
Edwards, B. H. (2011, June 16). Watch mathematics describing the real world: Precalculus and trigonometry [Video]. Prime Video. https://www.amazon.com/gp/video/detail/B08QLXN9NT/ref=atv_dp_share_cu_r
McKenzie-McHarg, A. (2020, February 19). Using DESMOS to graph radioactive decay [Video]. YouTube. https://www.youtube.com/watch?v=YdFCqy0BUv0&feature=youtu.be
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