To model the exponential decay of the radioactive substance, we can use the exponential decay formula:
\[ N(t) = N_0 \cdot e^{-kt} \]
where:
- \( N(t) \) is the amount of substance remaining after time \( t \),
- \( N_0 \) is the initial amount of the substance,
- \( k \) is the decay constant,
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm.
Given that the substance decays at a rate of 5% per year, we find the decay constant \( k \) as follows:
\[ k = 0.05 \]
So the model can be expressed as:
\[ N(t) = 500 \cdot e^{-0.05t} \]
Next, we need to calculate the amount of the substance remaining in the year 2000. The year 2000 is 10 years after 1990, so we set \( t = 10 \):
\[ N(10) = 500 \cdot e^{-0.05 \cdot 10} \]
Calculating the exponent:
\[ N(10) = 500 \cdot e^{-0.5} \]
Now, we compute \( e^{-0.5} \):
\[ e^{-0.5} \approx 0.6065 \]
Thus, we can continue with our calculation:
\[ N(10) = 500 \cdot 0.6065 \approx 303.25 \]
Therefore, the amount of the radioactive substance remaining in the year 2000 is approximately 303.25 grams.
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