Problem 23 - 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. $$ \begin{array}{|c|c|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Half-Life } \\ \text { (in years) } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years } \end{array} & \begin{array}{c} \text { Time Intervals, } \boldsymbol{x}\left(\frac{\text { Years }}{\text { Half-Life }}\right) \\ \text { Rounded to Tenths if Needed } \end{array} & \begin{array}{c} \text { Final Amount after } \boldsymbol{x} \text { Time } \\ \text { Intervals (rounded to tenths) } \end{array} & \begin{array}{c} \text { Is Your Final Amount } \\ \text { Reasonable? } \end{array} \\ \hline 60 & 8 & 10 & \frac{10}{8}=1.25 & 25.2 & \text { yes } \\ \hline \text { a. } 21 & 152 & 500 & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Final amount is 2.3, which is reasonable.

Step 1: Understand the Exponential Decay Formula

The exponential decay formula relating to half-life can be expressed as: \( A = A_0 \left(\frac{1}{2}\right)^x \), where \( A \) is the final amount, \( A_0 \) is the original amount, and \( x \) is the number of half-life time intervals elapsed.

Step 2: Calculate the Time Intervals

First, we need to calculate the number of half-life intervals for the given time frame. This is calculated using the formula: \( x = \frac{\text{Number of Years}}{\text{Half-Life}} \). For this problem: \( x = \frac{500}{152} \approx 3.29 \).

Step 3: Calculate the Final Amount

Using the number of time intervals calculated, plug into the decay formula to find the final amount: \( A = 21 \left(\frac{1}{2}\right)^{3.29} \). Simplifying this gives \( A \approx 2.3 \), which is the amount remaining after 3.29 half-lives.

Step 4: Verify the Reasonableness of the Final Amount

The calculated final amount is \( 2.3 \). Initially, you start with 21 and given the half-life and years, reaching \( 2.3 \) seems reasonable when you consider the decay over 500 years.

Key Concepts

Half-Life

The concept of half-life is an essential topic when discussing exponential decay in processes like radioactive decay or cooling. A half-life refers to the period it takes for a substance to reduce to half its initial quantity. Imagine you have a material that starts with a mass of 100 grams. After one half-life, only 50 grams remain. After another half-life, it further reduces to 25 grams, and so forth.

Understanding half-life is crucial because it helps predict how long a substance will last or the time it would take to reach a certain quantity. In scientific terms, it is a constant for each particular substance and remains unchanged over time. Importantly, half-life simplifies complex decay processes, allowing scientists and students to make accurate predictions about the future state of a material undergoing exponential decay.

When considering exercises like the one given, being able to calculate the number of half-lives that occur in a certain period is crucial to determining the amount of substance left over.

Exponential Decay Formula

The exponential decay formula is a powerful tool in mathematics and science, especially in understanding processes like radioactive decay and population decrease. It's expressed as: \[ A = A_0 \left(\frac{1}{2}\right)^x \]

• \(A\): the final amount left after decay over time\(x\)
• \(A_0\): the original starting amount
• \(\left(\frac{1}{2}\right)^x\): the decay factor, where \(x\) is the number of time intervals, like half-lives

This formula tells us how much of a material remains after a certain number of its half-life periods have occurred. Each half-life results in the material becoming half its previous amount, thus the expression \(\left(\frac{1}{2}\right)^x\) is used to signify that halving process.

Using the exponential decay formula, you can determine how much of a substance will be left after a given number of years. This is essential for scientists and engineers to predict the behavior of materials over time. Applying this in exercises, such as calculating remaining quantities of a substance over time, helps reinforce understanding and the practical applications of this concept.

Decay Over Time

Decay over time is a way to describe how materials or populations decrease in size or number as time progresses. In the context of exponential decay, decay over time follows a predictable pattern where the rate of decrease is proportional to the current amount.

Why is this concept important? It helps in modeling real-world scenarios, such as predicting how a radioactive substance will diminish or how quickly a drug metabolizes in the body. In the context of the exercise, understanding decay over time involves knowing how to calculate the number of half-lives that occur within a given period, as well as applying the exponential decay formula to find the final amount.

For instance, by dividing the number of years by the half-life of a material, you find how many times the material's quantity is halved. This informs calculations of the remaining amount: as we saw, 500 years with a half-life of 152 years gives approximately 3.29 intervals. By understanding decay over time, students can assess the reasonableness of the calculated remaining amount, making sense of how much is left after many intervals of decay.