To determine how much of the radioactive isotope will remain after 10 years, we can use the decay formula:
\[ N(t) = N_0 e^{-kt} \]
where:
- \(N(t)\) = remaining quantity of the substance after time \(t\),
- \(N_0\) = initial quantity,
- \(k\) = decay constant,
- \(t\) = time in years.
In this case, we know:
- \(N_0 = 200\) grams,
- \(k = 0.0278\) (decay rate),
- \(t = 10\) years.
First, we'll calculate \(N(10)\):
\[ N(10) = 200 e^{-0.0278 \cdot 10} \]
Calculating the exponent:
\[ -0.0278 \cdot 10 = -0.278 \]
Now, calculate \(e^{-0.278}\):
Using a calculator or a mathematical software to find \(e^{-0.278}\): \[ e^{-0.278} \approx 0.757 \]
Now, we can plug it back into the equation:
\[ N(10) \approx 200 \times 0.757 \]
Calculating this:
\[ N(10) \approx 151.46 \text{ grams} \]
Thus, the amount of the radioactive isotope that will remain after 10 years is approximately 151.46 grams.
Final Answer: There will be approximately 151.46 grams after 10 years.