Use the exponential decay model, [tex]A = A_0 e^{kt}[/tex], to solve the following:
The half-life of a certain substance is 30 years. How long will it take for a sample of this substance to decay to [tex]94\%[/tex] of its original amount?
Certainly! Let's solve the problem using the exponential decay model:
[tex]\[ A = A_0 e^{kt} \][/tex]
where: - [tex]\(A\)[/tex] is the amount of substance remaining after time [tex]\(t\)[/tex] - [tex]\(A_0\)[/tex] is the initial amount of the substance - [tex]\(k\)[/tex] is the decay constant - [tex]\(t\)[/tex] is the time
### Step-by-Step Solution:
#### Step 1: Understand the given information - Half-life ([tex]\(t_{1/2}\)[/tex]): The time it takes for half of the substance to decay, which is given as 30 years. - Decay to 94% of its original amount: This means [tex]\(A = 0.94 A_0\)[/tex].
#### Step 2: Relate the half-life to the decay constant [tex]\(k\)[/tex]
The decay constant [tex]\(k\)[/tex] is related to the half-life by the formula:
[tex]\[ k = \frac{\ln(0.5)}{t_{1/2}} \][/tex]
Substitute the given half-life (30 years):
[tex]\[ k = \frac{\ln(0.5)}{30} \][/tex]
#### Step 3: Calculate the value of [tex]\(k\)[/tex] [tex]\[ \ln(0.5) \approx -0.693 \][/tex]
Thus:
[tex]\[ k = \frac{-0.693}{30} \][/tex] [tex]\[ k \approx -0.023104906018664842 \][/tex]
#### Step 4: Set up the exponential decay equation
We know the substance decays to 94% of its original amount:
[tex]\[ A = 0.94 A_0 \][/tex]
Substitute [tex]\(A\)[/tex] and [tex]\(k\)[/tex] into the exponential decay equation:
