Problem 29 - 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 lead is 22 years. How long will it take for a sample of this substance to decay to \(80 \%\) of its original amount?

Step-by-Step Solution

Answer

It will take approximately 29.3 years for a sample of lead to decay to 80% of its original amount.

Step 1: Understand the decay model

The decay model is given by the equation \(A=A_{0} e^{k t}\), where \(A\) is the final amount, \(A_{0}\) is the initial amount, \(k\) is the decay rate, and \(t\) is the time. For simplicity, let's denote the initial amount as 1 (100%), so we want to find the time it takes to decay to 0.8 (80%)

Step 2: Substitute half-life into the decay equation to find \(k\)

Knowing that the half-life of lead is 22 years (i.e it would decay to 50% or 0.5 of its original amount in that time), replace \(A\) with 0.5 and \(t\) with 22 in the decay equation to find \(k\). This would give: \(0.5= e^{22k}\). Take the natural log of both sides and solve for \(k\). Thus, \(k\) would equal to \(\frac{ln(0.5)}{22}\)

Step 3: Substitute the values into the decay equation to find the required time

Substitute \(A\) with 0.8 (80% we are required to calculate for) and \(k\) with the computed value in step 2 into the decay equation : \(0.8 = e^{k t}\). Take the natural log of both sides and solve for \(t\). Thus, \(t\) would equal to \(\frac{ln(0.8)}{k}\), where \(k\) is the decay rate obtained in step 2.