A certain radioactive substance decays at a rate of 5% per year. In 1990, the initial amount of the substance was 500 grams. Write an exponential decay function to model the decay of the substance over time. Calculate the amount of the substance remaining in 2000.

1 answer

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.