- Identify the parameters: initial amount \(A = 10\text{ mg}\), half-life \(h = 8\text{ days}\), and time \(t = 16\text{ days}\).
- Use the decay formula \(P(t) = A \left(\frac{1}{2}\right)^{\frac{t}{h}}\).
- Substitute the values to find \(P(16) = 10 \cdot \left(\frac{1}{2}\right)^{\frac{16}{8}} = 10 \cdot \left(\frac{1}{2}\right)^2\).
- Calculate the final value, which is \(\boxed{2.5\text{ mg}}\).
Explanation
- Problem Analysis
We are given that the radioactive decay of a sample follows the equation \(P(t) = A \left(\frac{1}{2}\right)^{\frac{t}{h}}\), where:
- \(P(t)\) is the amount remaining after time \(t\).
- \(A\) is the initial amount of the sample.
- \(h\) is the half-life of the isotope.
- \(t\) is the time elapsed.
From the problem statement, we have:
- Initial amount \(A = 10\text{ mg}\).
- Half-life \(h = 8\text{ days}\).
- Time elapsed \(t = 16\text{ days}\).
We need to calculate \(P(16)\) using these values.
- Calculating the Remaining Amount
To find the amount remaining after 16 days, we substitute the known values into the decay equation:
\[P(16) = 10 \cdot \left(\frac{1}{2}\right)^{\frac{16}{8}}\]
First, calculate the exponent:
\[\frac{16}{8} = 2\]
So the equation simplifies to:
\[P(16) = 10 \cdot \left(\frac{1}{2}\right)^2\]
Now, evaluate \(\left(\frac{1}{2}\right)^2\):
\[\left(\frac{1}{2}\right)^2 = \frac{1}{4} = 0.25\]
Finally, multiply this by the initial amount \(A = 10\):
\[P(16) = 10 \cdot 0.25 = 2.5\text{ mg}\]
- Conclusion
Based on our calculation, the remaining amount of the radioactive isotope after 16 days is \(2.5\text{ mg}\). This corresponds to option (b) in the provided list. Therefore, the correct amount is \(\boxed{2.5\text{ mg}}\).
Examples
Radioactive decay models are fundamental in fields like archaeology and medicine. For example, archaeologists use carbon-14 dating to determine the age of ancient organic materials. Since they know the half-life of carbon-14, they can measure the remaining amount of the isotope in a sample to calculate how many thousands of years have passed since the organism died. This same logic helps medical professionals calculate safe dosages for radioactive tracers used in diagnostic imaging, ensuring that the radioactive material decays to a harmless level within the body.