Problem 24 - Algebra A Combined Function

Question

Practice using the exponential decay formula with half-lives by completing the table below. The first row has been completed for you. $$ \begin{array}{|c|c|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Half-Life } \\ \text { (in years) } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years } \end{array} & \begin{array}{c} \text { Time Intervals, } \boldsymbol{x}\left(\frac{\text { Years }}{\text { Half-Life }}\right) \\ \text { Rounded to Tenths if Needed } \end{array} & \begin{array}{c} \text { Final Amount after } \boldsymbol{x} \text { Time } \\ \text { Intervals (rounded to tenths) } \end{array} & \begin{array}{c} \text { Is Your Final Amount } \\ \text { Reasonable? } \end{array} \\ \hline 60 & 8 & 10 & \frac{10}{8}=1.25 & 25.2 & \text { yes } \\ \hline \text { a. } 35 & 119 & 500 & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Final amount after 500 years is 1.3; calculation is reasonable.

Step 1: Understand the Exponential Decay Formula

The formula for calculating the final amount via exponential decay given a half-life is: \( A = A_0 \left(\frac{1}{2}\right)^x \), where \( A_0 \) is the original amount, and \( x \) is the number of half-life time intervals, calculated as \( x = \frac{\text{Number of Years}}{\text{Half-Life}} \).

Step 2: Calculate the Number of Time Intervals

Given the problem, you have an original amount of 35, a half-life of 119 years, and a time span of 500 years. First, calculate the number of half-life intervals: \( x = \frac{500}{119} \approx 4.2 \).

Step 3: Apply the Exponential Decay Formula

Substitute the values into the exponential decay formula: \[ A = 35 \left(\frac{1}{2}\right)^{4.2} \]. This will give you the final amount after 500 years.

Step 4: Calculate the Final Amount

Compute \( A = 35 \left(\frac{1}{2}\right)^{4.2} \) using a calculator: \[ A \approx 1.3 \]. Thus, the final amount after 500 years is approximately 1.3.

Step 5: Evaluating Reasonableness

With an original amount of 35 and a long period of 500 years (over 4 half-lives), it is reasonable that the final amount is significantly lower, approximately 1.3.

Key Concepts

Half-life

Half-life is a fundamental concept in understanding exponential decay processes, such as radioactive decay and certain chemical reactions.
The half-life of a substance is the time required for half of the original quantity to decay or transform into another state.

This doesn't just mean half of the substance disappears; rather, it changes its form. - The concept of half-life is independent of the initial amount, meaning that everything (regardless of starting quantity) decays by half in the given time. - It is also a constant period unique to each substance, and it usually remains unchanged over the substance's lifetime. For instance, if a material has a half-life of 10 years and you start with 100 grams, after 10 years, you will have 50 grams left.
After 20 years, you will have 25 grams, and so on.

Time Intervals

Time intervals are an essential aspect of using the half-life concept in calculations. To determine how much of a substance is left after a certain period, you first need to calculate how many half-lives fit into that period.
For this, you divide the total time by the half-life:- Formula: \( x = \frac{\text{Total Time}}{\text{Half-Life}} \)Let's use a practical example: if the half-life of a material is 119 years, and you want to know what happens in 500 years:- Divide 500 by 119, which calculates to approximately 4.2.- This value means 500 years is roughly 4.2 half-life periods.By knowing the number of time intervals, you can determine how the substance has decayed over the chosen period.

Exponential Decay Formula

The exponential decay formula is vital for predicting how a quantity diminishes over time when its rate of decrease is proportional to its present value. This is common in processes with a half-life.
The formula is expressed as:\[ A = A_0 \left(\frac{1}{2}\right)^x \]- \( A_0 \) is the initial or original amount.- \( x \) is the number of half-life time intervals, calculated as described earlier.For example, using the exponential decay formula to calculate how much of an initial 35 units remain after 4.2 half-life periods:- Insert into the formula: \( A = 35 \left(\frac{1}{2}\right)^{4.2} \)- This will give the amount remaining after 500 years.By using this formula, you can find out the remaining quantity of the original amount after any number of time intervals.

Final Amount Calculation

Final amount calculation is the concluding step to determine how much of a substance remains after applying the exponential decay model.- To find the final amount, use a calculator to perform the power of division operations in the formula: \( A = A_0 \left(\frac{1}{2}\right)^x \).With the given example, where the initial amount \( A_0 \) is 35, and \( x \), the time intervals, is 4.2:- You calculate: \( A = 35 \left(\frac{1}{2}\right)^{4.2} \), resulting in approximately 1.3.This tells us that after 500 years, only about 1.3 units remain out of the original 35.
Remember, the reasonableness of the final value should be checked against expectations. In this scenario, a drastically reduced amount is expected over the long span of 500 years, confirming the calculation's reliability.