The half-life of a certain radioactive material is 78 hours. An initial amount of the material has a mass of 790 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 18 hours. Round your answer to the nearest thousandth.
The exponential decay function is given by
\[N(t) = N_0 \cdot e^{-\lambda t}\]
where:
- \(N(t)\) is the amount of radioactive material remaining after time \(t\),
- \(N_0\) is the initial amount of radioactive material,
- \(\lambda\) is the decay constant (related to the half-life by \(\lambda = \frac{{\ln(2)}}{{\text{{half-life}}}}\)), and
- \(t\) is the time in hours.
Given that the half-life is 78 hours, we can calculate \(\lambda\):
\[\lambda = \frac{{\ln(2)}}{{78}} \approx 0.0088976\]
Then we can write the exponential decay function as:
\[N(t) = 790 \cdot e^{-0.0088976t}\]
To find the amount of radioactive material remaining after 18 hours, we can substitute \(t = 18\) into the equation:
\[N(18) = 790 \cdot e^{-0.0088976(18)} \approx 708.799\]
Rounding to the nearest thousandth, the amount of radioactive material remaining after 18 hours is approximately 708.799 kg.