To determine the time it will take for a 50 mg sample of a radioactive element to decay to 10 mg given the exponential decay model [tex]\( P(t) = P_0 e^{-0.004261t} \)[/tex], follow these steps:
1. Identify the given quantities:
- Initial amount ([tex]\( P_0 \)[/tex]): 50 mg
- Final amount ([tex]\( P(t) \)[/tex]): 10 mg
- Decay constant ([tex]\( k \)[/tex]): 0.004261
2. Set up the equation:
The formula for radioactive decay is:
[tex]\[
P(t) = P_0 e^{-kt}
\][/tex]
Substituting the given values, we get:
[tex]\[
10 = 50 e^{-0.004261t}
\][/tex]
3. Isolate the exponential term:
Divide both sides of the equation by 50 to isolate the exponential expression:
[tex]\[
\frac{10}{50} = e^{-0.004261t}
\][/tex]
Simplify the fraction:
[tex]\[
0.2 = e^{-0.004261t}
\][/tex]
4. Take the natural logarithm of both sides:
Applying the natural logarithm [tex]\( \ln \)[/tex] to both sides of the equation allows us to solve for [tex]\( t \)[/tex]:
[tex]\[
\ln(0.2) = \ln(e^{-0.004261t})
\][/tex]
Using the property of logarithms [tex]\( \ln(e^x) = x \)[/tex], we get:
[tex]\[
\ln(0.2) = -0.004261t
\][/tex]
5. Solve for [tex]\( t \)[/tex]:
Rearrange the equation to solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln(0.2)}{-0.004261}
\][/tex]
6. Calculate [tex]\( \ln(0.2) \)[/tex] and then [tex]\( t \)[/tex]:
Compute [tex]\( t \)[/tex] using the given decay constant:
[tex]\[
t = \frac{\ln(0.2)}{-0.004261}
\][/tex]
By solving this, we get:
[tex]\[
t \approx 377.71366168366586
\][/tex]
7. Round the final answer to one decimal place:
[tex]\[
t \approx 377.7
\][/tex]
Hence, it will take approximately 377.7 days for a 50 mg sample of the radioactive element to decay to 10 mg.
