The amount of a radioactive isotope remaining after a certain time can be calculated using the formula:
\[ N(t) = N_0 \times e^{-kt} \]
Where:
- \( N(t) \) is the remaining quantity of the substance after time \( t \).
- \( N_0 \) is the initial quantity of the substance.
- \( k \) is the decay constant.
- \( t \) is the time elapsed.
- \( e \) is the base of the natural logarithm.
From the problem, we have:
- \( N_0 = 200 \) grams
- The decay rate \( k = 0.0278 \)
- \( t = 10 \) years
Now we can substitute the values into the formula:
\[ N(10) = 200 \times e^{-0.0278 \times 10} \]
Calculating \( -0.0278 \times 10 \):
\[ -0.0278 \times 10 = -0.278 \]
Now we calculate \( e^{-0.278} \):
\[ e^{-0.278} \approx 0.757 (using a calculator or mathematical tables) \]
Now substituting this back into the equation:
\[ N(10) \approx 200 \times 0.757 \] \[ N(10) \approx 151.4 \text{ grams} \]
Therefore, after 10 years, approximately 151.4 grams of the radioactive isotope will remain.