Question

Radioactive Decay: A certain radioactive material decays so that after each year the radioactivity is \(8 \%\) less than at the start of that year. How many years will it take for its radioactivity to be \(50 \%\) of its original value?

Short answer

Using the formula \(t = \frac{\log(0.5)}{\log(0.92)}\), it takes approximately 9 years for the radioactivity to be 50% of its original value.

Step-by-step solution

Understand the Decay Process

The decay process described is exponential, which can be modeled by the formula \(A(t) = A_0(1 - r)^t\), where \(A(t)\) is the amount of radioactivity at time \(t\), \(A_0\) is the initial amount of radioactivity, \(r\) is the decay rate, and \(t\) is the time in years.

Set Up the Decay Equation

We set up the decay equation as follows: \(0.5A_0 = A_0(1 - 0.08)^t\). The \(0.5A_0\) represents 50% of the original radioactivity, and \(1 - 0.08\) is the annual decay factor.

Simplify and Solve for t

Simplify the equation to \(0.5 = (0.92)^t\). To solve for \(t\), take the logarithm of both sides and solve for \(t\) using the properties of logarithms: \(\log(0.5) = t\log(0.92)\). Dividing both sides by \(\log(0.92)\) gives us \(t = \frac{\log(0.5)}{\log(0.92)}\). Using a calculator, we find \(t\).

Calculate the Number of Years

After finding the value of \(t\), round up if necessary, because you cannot have a fraction of a year in the context of this decay scenario.

Key concepts

Exponential Decay

When discussing radioactive decay, we encounter an important mathematical model known as exponential decay. This process is characterised by a consistent percentage reduction in the remaining quantity of the substance over equal time intervals. In other words, rather than decreasing by a fixed amount each period, the quantity decreases by a fixed percentage, leading to a rapid decrease initially that slows down as time progresses.

To visualize this, imagine a slice of cake that gets 8% smaller each time you take a bite, regardless of its current size. With each bite, the actual amount you eat becomes less because each bite is only 8% of what's left, not of the original whole cake. This is similar to how radioactive materials lose their radioactivity over time.

Mathematically, we can express exponential decay with the function \(A(t) = A_0(1 - r)^t\), where \(A_0\) represents the initial amount, \(r\) the decay rate (in this case, 8% or 0.08), and \(t\) time, usually in periods like years or days. This model is powerful because it describes a broad range of decay processes found in nature, including the decay of radioactive substances.

Decay Rate

The decay rate in an exponential decay model is crucial as it determines how quickly the quantity decreases over time. It is represented by the variable \(r\) in the decay equation \(A(t) = A_0(1 - r)^t\), and it is always a positive number between 0 and 1. This number reflects the percentage of the substance that decays in one standard interval of time, such as a year in our example.

For radioactive substances, the decay rate is inherent to the material and can be influenced by various factors, including the type of nuclear reaction occurring. It's also important to note that while the decay rate is constant, the actual amount of substance that decays in each period will decrease, as it's a percentage of the remaining substance.

The concept of the decay rate helps scientists predict the behavior of radioactive materials and is essential for applications such as dating archaeological finds, understanding nuclear power, and managing medical treatments involving radioactive substances. By controlling the variables in the exponential decay equation, you can predict when a substance will fall to a certain level of radioactivity, which is exactly what we do when we want to know how long it will take for a substance to decay to half of its original amount.

Logarithmic Equations

In the solution of our original problem, logarithmic equations play a pivotal role in determining the time required for the radioactive material to decay to half its initial value. Logarithms are the inverse operations of exponentiation, meaning they allow us to 'unpack' exponential relationships and solve for unknowns such as time in our exponential decay model.

To solve the equation \(0.5 = (0.92)^t\), we take the logarithm of both sides, leading to \(\frac{\text{log}(0.5)}{\text{log}(0.92)} = t\), where \(t\) is the time in years for the radioactivity to be reduced by 50%. Through the properties of logarithms, we can extract the exponent \(t\) and isolate it on one side of the equation.

This technique comes in handy not only in theoretical exercises but also in practical applications. Whether it's calculating the half-life of a radioactive isotope or determining the time it will take for an investment to grow to a certain amount under continuous compounding interest, the ability to work with logarithms is a critical tool in the mathematics of exponential processes.