- The decay of a radioactive substance is modeled by the formula \(A(t) = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{h}}\).
- Substituting the given values \(A_0 = 1000\), \(h = 15\), and \(t = 45\) gives the expression \(A(45) = 1000 \times \left(\frac{1}{2}\right)^{\frac{45}{15}}\).
- Simplifying the exponent results in \(A(45) = 1000 \times \left(\frac{1}{2}\right)^3 = 1000 \times \frac{1}{8} = 125\).
- Since the calculated amount is 125 grams, the conclusion that 125 grams of sodium-24 will remain is \(\boxed{\text{True}}\).
Explanation
- Analysis of the Radioactive Decay Problem
We are given an initial sample of 1,000 grams of radioactive sodium-24. We need to determine the amount of sodium-24 remaining after 45 hours, given that its half-life is 15 hours. The formula for radioactive decay is given by \[A(t) = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{h}}\] where \(A(t)\) is the remaining amount, \(A_0\) is the initial amount, \(t\) is the time elapsed, and \(h\) is the half-life.
- Setting up the Calculation
Using the values provided:
- Initial amount (\(A_0\)) = 1,000 grams
- Half-life (\(h\)) = 15 hours
- Elapsed time (\(t\)) = 45 hours
We substitute these values into the radioactive decay formula:
\[A(45) = 1000 \times \left(\frac{1}{2}\right)^{\frac{45}{15}}\]
- Calculating the Remaining Amount and Conclusion
First, we simplify the exponent \(\frac{45}{15} = 3\).
Now, we calculate the remaining amount:
\[A(45) = 1000 \times \left(\frac{1}{2}\right)^3\]
\[A(45) = 1000 \times \frac{1}{8}\]
\[A(45) = 125\]
The amount of sodium-24 remaining after 45 hours is 125 grams, which confirms the conclusion is correct.
Examples
Radioactive decay calculations are essential in fields such as archaeology and medicine. For instance, carbon-14 dating helps scientists determine the age of ancient artifacts by measuring how much of the radioactive carbon has decayed over time. Similarly, in nuclear medicine, doctors use radioactive tracers with specific half-lives to diagnose conditions, ensuring the material remains active for long enough to be useful while decaying quickly enough to be safe for the patient. Understanding this decay model helps in managing these processes accurately.