Problem 20 - Precalculus Mathematics for Calculus

Question

These exercises use the radioactive decay model. Radium-221 has a half-life of 30 s. How long will it take for \(95 \%\) of a sample to decay?

Step-by-Step Solution

Answer

It takes approximately 129.90 seconds for 95% of the sample to decay.

Step 1: Understand Half-Life

The half-life of a substance is the time it takes for half of the substance to decay. For Radium-221, this time is 30 seconds.

Step 2: Set Up the Decay Formula

The decay of a substance over time is modeled by the formula \( N(t) = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \), where \( N_0 \) is the initial amount, \( N(t) \) is the amount remaining after time \( t \), and \( T_{1/2} \) is the half-life.

Step 3: Apply Example Values in the Formula

Given that 95% of the sample decays, 5% remains. Therefore, \( N(t) = 0.05N_0 \). Substitute this into the decay formula: \( 0.05N_0 = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{30}} \).

Step 4: Simplify the Equation

Cancel \( N_0 \) from both sides of the equation: \( 0.05 = \left( \frac{1}{2} \right)^{\frac{t}{30}} \).

Step 5: Solve for t Using Logarithms

Take the natural logarithm on both sides: \( \ln(0.05) = \ln\left( \left( \frac{1}{2} \right)^{\frac{t}{30}} \right) \). This simplifies to \( \ln(0.05) = \frac{t}{30} \cdot \ln(0.5) \).

Step 6: Calculate t

Isolate \( t \) by multiplying both sides by 30 and dividing by \( \ln(0.5) \): \( t = \frac{30 \cdot \ln(0.05)}{\ln(0.5)} \). Using a calculator, \( \ln(0.05) \approx -2.9957 \) and \( \ln(0.5) \approx -0.6931 \). Substituting these values gives \( t \approx 129.90 \) seconds.

Key Concepts

Half-Life

Half-life is a fundamental concept in radioactive decay. It refers to the time it takes for half of a radioactive substance to decay into another substance. This period is constant for any given substance. In our example, Radium-221 has a half-life of 30 seconds. This means if you start with a certain amount of Radium-221, half of it will have decayed after 30 seconds.

Half-life helps us understand how quickly a substance will decay and is essential in fields such as archaeology and biology. For every next half-life interval, another half of the remaining substance decays, but never the entire amount. So, in successive half-lives, the amount continues to halve.

Decay Formula

The decay formula is a mathematical model used to describe the process of radioactive decay over time. It is expressed as:

\[ N(t) = N_0 \cdot \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]

where:

• \(N(t)\) is the quantity remaining after time \(t\).
• \(N_0\) is the initial quantity of the substance.
• \(T_{1/2}\) is the half-life of the substance.

The decay formula enables scientists to predict how much of a substance will remain after a specific period.

In the exercise, if 95% of a sample has decayed, then 5% remains. Setting up the equation with this information allows us to solve for the time it will take for this decay to occur.

Natural Logarithms

Natural logarithms are often used to solve exponential decay equations like the radioactive decay formula. The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), which is approximately 2.718.

In the solution, we reached the equation:
\[ 0.05 = \left( \frac{1}{2} \right)^{\frac{t}{30}} \]
To solve for \(t\), we take the natural logarithm of both sides:

\[ \ln(0.05) = \ln\left( \left( \frac{1}{2} \right)^{\frac{t}{30}} \right) \]
This simplifies further using the property of logarithms that \(\ln(a^b) = b \cdot \ln(a)\):

\[ \ln(0.05) = \frac{t}{30} \cdot \ln(0.5) \]
By rearranging this equation and solving for \(t\), we can find how long it takes for 95% of the substance to decay. This step highlights the power of natural logarithms in breaking down exponential functions into linear ones, making complex calculations manageable.