Problem 21 - Precalculus with Limits

Question

complete the table for the radioactive isotope. Complete the table assuming continuously compounded interest. $$ \begin{array}{llll} \text { Isotope} & \text { Half-Life } & \text { Initial Quantity } & \text { Amount After } \\ \underline{\phantom{xxx}} & \ \text { Years } & \underline{\phantom{xxx}} & \text { 1000 Years } \\ \ ^{226}Ra&\quad1599 & \quad \space &\quad \\ \end{array} $$

Step-by-Step Solution

Answer

The decay constant for the isotope is approximately -0.000433 per year. The initial quantity is set as 1 unit for simplicity. After 1,000 years, approximately 0.646 units of the isotope remain.

Step 1: Determine the decay constant

First, find the decay constant (r) using the half-life of the isotope. Using the formula \( r = -ln(2)/half-life \), and substituting 1599 years for the half-life, you get \( r = -ln(2)/1599 = -0.000433 \).

Step 2: Set initial quantity

The initial quantity isn't given, so let's set it as an arbitrary value, let's say \( P = 1 \) for simplicity. Thus, initially you have one unit of the isotope.

Step 3: Calculate the quantity after 1,000 years

Then, you can calculate the amount of isotope left after 1,000 years using the formula for continuously compounded decay \( A = P*e^(rt) \). Substituting the values in, \( A = 1*e^(-0.000433*1000) \), hence \( A = 0.646 \). Therefore, after 1,000 years, approximately 0.646 units of the isotope remains.

Key Concepts

Continuously Compounded Interest

Understanding continuously compounded interest is critical for various applications, from finance to radioactive decay. In finance, it's observed when the interest earned on an investment is reinvested and generates its own earnings, which also accumulate interest, making the growth of the investment an exponential function of time. The formula to calculate the future value of an investment, using continuously compounded interest, is given by
\[ A = Pe^{rt} \]
where:\[A\] is the amount of money after a certain time, \[P\] is the principal amount, \[r\] is the interest rate, and \[t\] is the time. This approach is especially useful in our problem solving as we model the decay of a radioactive isotope very similarly to how money grows under continuous compounding.

Half-Life

Half-life is a term widely used in nuclear physics and chemistry to describe the time required for one-half of a radioactive substance to decay. This concept is critical for understanding radioactive decay processes. The half-life of a substance does not depend on its amount but purely on its inherent stability and the decay process it undergoes.

So, when we talk about the half-life in terms of a problem, we can use it to find out how much of a radioactive isotope will remain after a certain period. For instance, if a substance has a half-life of 5 years, after 5 years, only 50% of the initial amount would remain; after 10 years, 25%, and so on. This concept is fundamental for determining the decay constant as well.

Decay Constant

The decay constant, usually denoted by \(\lambda\) or \(r\) in some contexts, represents the probability of a single atom decaying per unit time. It is a crucial part of the equation for predicting the rate at which a substance will decay over time. The decay constant is directly related to the half-life of a substance as it dictates how quickly the substance transforms.

In the context of the exercise, we calculate the decay constant using the formula:
\[ r = \frac{-\ln(2)}{half-life} \]
In a simple term, the decay constant tells us the 'speed' of the decay. Knowing this constant allows us to apply the principles of continuously compounded interest to model the decay of isotopes.

Exponential Decay

Exponential decay is a pattern of decrease at a rate that is proportional to the amount present. This concept can be visualized in processes where the quantity decreases over time at a rate that depends on its current value. In other words, the larger the quantity, the faster the rate of decay. It's the exact opposite process of exponential growth, such as continuously compounded interest.

In the solved problem, we saw how the formula \[ A = Pe^{rt} \] applied, with a negative rate \(r\) indicating decay. Despite the context most commonly being finance for compounding interest, this mathematical model is also beautifully transferable to physical applications like the decay of radioactive isotopes. Exponential decay functions reveal that radioactive substances decay at a rate which decreases over time, conforming to their specific decay constants and half-lives.