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Answer :
Let's solve the problem step by step.
### a. Exponential Decay Model
The exponential decay model is given by the formula:
[tex]\[ A(t) = A_0 \cdot e^{(-kt)} \][/tex]
where:
- [tex]\( A(t) \)[/tex] is the amount of substance left at time [tex]\( t \)[/tex],
- [tex]\( A_0 \)[/tex] is the initial amount of the substance,
- [tex]\( e \)[/tex] is the base of the natural logarithm,
- [tex]\( k \)[/tex] is the decay constant,
- [tex]\( t \)[/tex] is time.
For a substance with a half-life of 38.8 years, we first need to find the decay constant [tex]\( k \)[/tex]. The formula to find the decay constant when the half-life, [tex]\( T_{1/2} \)[/tex], is known is:
[tex]\[ k = \frac{\ln(2)}{T_{1/2}} \][/tex]
Plugging in the given half-life:
[tex]\[ k = \frac{\ln(2)}{38.8} \approx 0.018 \][/tex]
Therefore, the exponential decay model is:
[tex]\[ A(t) = A_0 \cdot e^{(-0.018t)} \][/tex]
### b. Time to Decay to 300 Grams
We want to find how long it takes for a 400-gram sample to decay to 300 grams. Using the decay formula:
[tex]\[ 300 = 400 \cdot e^{(-0.018t)} \][/tex]
First, divide both sides by 400:
[tex]\[ \frac{300}{400} = e^{(-0.018t)} \][/tex]
[tex]\[ 0.75 = e^{(-0.018t)} \][/tex]
Take the natural logarithm of both sides:
[tex]\[ \ln(0.75) = -0.018t \][/tex]
Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(0.75)}{-0.018} \approx 16.103 \][/tex]
So, it will take approximately 16.1 years for the sample to decay to 300 grams.
### c. Amount Remaining After 10 Years
To find out how much of the 400-gram sample will remain after 10 years, use the exponential decay formula again:
[tex]\[ A(10) = 400 \cdot e^{(-0.018 \times 10)} \][/tex]
Calculate:
[tex]\[ A(10) \approx 334.561 \][/tex]
So, approximately 334.56 grams of the sample will remain after 10 years.
### a. Exponential Decay Model
The exponential decay model is given by the formula:
[tex]\[ A(t) = A_0 \cdot e^{(-kt)} \][/tex]
where:
- [tex]\( A(t) \)[/tex] is the amount of substance left at time [tex]\( t \)[/tex],
- [tex]\( A_0 \)[/tex] is the initial amount of the substance,
- [tex]\( e \)[/tex] is the base of the natural logarithm,
- [tex]\( k \)[/tex] is the decay constant,
- [tex]\( t \)[/tex] is time.
For a substance with a half-life of 38.8 years, we first need to find the decay constant [tex]\( k \)[/tex]. The formula to find the decay constant when the half-life, [tex]\( T_{1/2} \)[/tex], is known is:
[tex]\[ k = \frac{\ln(2)}{T_{1/2}} \][/tex]
Plugging in the given half-life:
[tex]\[ k = \frac{\ln(2)}{38.8} \approx 0.018 \][/tex]
Therefore, the exponential decay model is:
[tex]\[ A(t) = A_0 \cdot e^{(-0.018t)} \][/tex]
### b. Time to Decay to 300 Grams
We want to find how long it takes for a 400-gram sample to decay to 300 grams. Using the decay formula:
[tex]\[ 300 = 400 \cdot e^{(-0.018t)} \][/tex]
First, divide both sides by 400:
[tex]\[ \frac{300}{400} = e^{(-0.018t)} \][/tex]
[tex]\[ 0.75 = e^{(-0.018t)} \][/tex]
Take the natural logarithm of both sides:
[tex]\[ \ln(0.75) = -0.018t \][/tex]
Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(0.75)}{-0.018} \approx 16.103 \][/tex]
So, it will take approximately 16.1 years for the sample to decay to 300 grams.
### c. Amount Remaining After 10 Years
To find out how much of the 400-gram sample will remain after 10 years, use the exponential decay formula again:
[tex]\[ A(10) = 400 \cdot e^{(-0.018 \times 10)} \][/tex]
Calculate:
[tex]\[ A(10) \approx 334.561 \][/tex]
So, approximately 334.56 grams of the sample will remain after 10 years.
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