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 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.