The general formula for exponential decay is given by:
\[A(t) = A_0 \cdot e^{kt}\]
where:
- A(t) represents the amount of material at time t
- A_0 represents the initial amount of material
- k is the decay constant
- t is the time elapsed
We can find the decay constant k using the formula:
\[k = \frac{ln(0.5)}{t_{1/2}}\]
where t_{1/2} is the half-life.
Substituting the given values: t_{1/2} = 78 hours and A_0 = 790 kg, we can calculate k:
\[k = \frac{ln(0.5)}{78} \approx -0.00889\]
Now, we can write the exponential function that models the decay:
\[A(t) = 790 \cdot e^{-0.00889t}\]
To find the amount of radioactive material remaining after 18 hours, we substitute t = 18 into the equation:
\[A(18) = 790 \cdot e^{-0.00889(18)} \approx 696.282\]
Rounded to the nearest thousandth, approximately 696.282 kg of radioactive material remains after 18 hours.