The half-life of a certain radioactive material is 42 days. An initial amount of the material has a mass of 49 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 8 days. Round your answer to the nearest thousandth.
The exponential function that models the decay of the material can be written as:
\[y = A \left(\frac{1}{2}\right)^{\frac{t}{h}}\]
where:
- $y$ represents the remaining mass of the material after time $t$.
- $A$ represents the initial mass of the material.
- $h$ represents the half-life of the material.
Plugging in the given values, we have:
\[y = 49 \left(\frac{1}{2}\right)^{\frac{t}{42}}\]
To find how much radioactive material remains after 8 days, we substitute $t = 8$ into the function:
\[y = 49 \left(\frac{1}{2}\right)^{\frac{8}{42}}\]
\[y \approx 49 \cdot (0.792)\]
\[y \approx 38.808\]
Rounded to the nearest thousandth, the amount of radioactive material remaining after 8 days is approximately \boxed{38.808\text{ kg}}.