Answer :
To find the exponential decay model for a radioactive substance with a given half-life, we can follow these steps:
1. Understand the Concept of Half-Life:
The half-life of a substance is the time it takes for half of the substance to decay. For this problem, the half-life is given as 39.3 years.
2. Identify the Formula for Exponential Decay:
The general formula for exponential decay is:
[tex]\[
A(t) = A_0 \cdot e^{kt}
\][/tex]
Where:
- [tex]\(A(t)\)[/tex] is the amount of the 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,
- [tex]\(e\)[/tex] is the base of the natural logarithm.
3. Calculate the Decay Constant ([tex]\(k\)[/tex]):
The decay constant is crucial for modeling the decay process and can be calculated from the half-life using the formula:
[tex]\[
k = \frac{\ln(2)}{\text{half-life}}
\][/tex]
Here, [tex]\(\ln(2)\)[/tex] is the natural logarithm of 2.
4. Substitute the Given Half-Life:
Plug the given half-life of 39.3 years into the decay constant formula:
[tex]\[
k = \frac{\ln(2)}{39.3}
\][/tex]
5. Compute the Decay Constant:
The decay constant [tex]\(k\)[/tex] is approximately 0.018 when rounded to the nearest thousandth.
6. Form the Exponential Decay Model:
With the decay constant determined, you can express the exponential decay model as:
[tex]\[
A(t) = A_0 \cdot e^{-0.018t}
\][/tex]
This equation can now be used to model the decay of the substance over time, where [tex]\(A_0\)[/tex] is the initial amount.
This model allows you to predict how much of the radioactive substance will remain after a given period.
1. Understand the Concept of Half-Life:
The half-life of a substance is the time it takes for half of the substance to decay. For this problem, the half-life is given as 39.3 years.
2. Identify the Formula for Exponential Decay:
The general formula for exponential decay is:
[tex]\[
A(t) = A_0 \cdot e^{kt}
\][/tex]
Where:
- [tex]\(A(t)\)[/tex] is the amount of the 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,
- [tex]\(e\)[/tex] is the base of the natural logarithm.
3. Calculate the Decay Constant ([tex]\(k\)[/tex]):
The decay constant is crucial for modeling the decay process and can be calculated from the half-life using the formula:
[tex]\[
k = \frac{\ln(2)}{\text{half-life}}
\][/tex]
Here, [tex]\(\ln(2)\)[/tex] is the natural logarithm of 2.
4. Substitute the Given Half-Life:
Plug the given half-life of 39.3 years into the decay constant formula:
[tex]\[
k = \frac{\ln(2)}{39.3}
\][/tex]
5. Compute the Decay Constant:
The decay constant [tex]\(k\)[/tex] is approximately 0.018 when rounded to the nearest thousandth.
6. Form the Exponential Decay Model:
With the decay constant determined, you can express the exponential decay model as:
[tex]\[
A(t) = A_0 \cdot e^{-0.018t}
\][/tex]
This equation can now be used to model the decay of the substance over time, where [tex]\(A_0\)[/tex] is the initial amount.
This model allows you to predict how much of the radioactive substance will remain after a given period.