Problem 28 - College Algebra Essentials

Question

Use the exponential decay model, \(A=A_{0} e^{k t},\) to solve Exercises \(28-31 .\) Round answers to one decimal place. The half-life of thorium-229 is 7340 years. How long will it take for a sample of this substance to decay to \(20 \%\) of its original amount?

Step-by-Step Solution

Answer

It will take approximately 18660.2 years for a sample of Thorium-229 to decay to 20 % of its original quantity.

Step 1: Understanding the Exponential Decay Model

The formula \(A=A_{0} e^{k t}\) represents exponential decay. In this formula, \(A_{0}\) is the original amount, \(A\) is the remaining amount after time \(t\), \(e\) is the base of natural logarithms, \(k\) is the decay constant and \(t\) is time.

Step 2: Calculating the Decay Constant from Half-life

From the half-life, \(T_{1/2} = 7340\) years, the decay constant \(k\) can be calculated using the formula \(k = -\frac{ln(2)}{T_{1/2}}\). Substituting the given values gives \(k = -\frac{ln(2)}{7340} = -9.44 \times 10^{-5} \) per year.

Step 3: Calculate the time to decay to 20% of the original amount

In order to find the time taken to decay to 20% of its original amount, we substitute these values into our decay formula to get \(0.2 = e^{k t}\). Take the natural logarithm of both sides to simplify the equation as follows: \(ln(0.2) = k t\). Solving for \(t\) gives \(t = \frac{ln(0.2)}{k} = 18660.2 \) years.