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.
(1 point)
We can use the formula for exponential decay, which is given by:
N(t) = N₀ * e^(-kt)
where N(t) is the remaining amount of radioactive material at time t, N₀ is the initial amount of the material, e is the base of the natural logarithm (approximately 2.71828), and k is the decay constant.
In this case, N₀ = 49 kg and the half-life is 42 days. We can use the half-life to find the decay constant, k.
Since the half-life is the time it takes for half of the radioactive material to decay, we have:
N(t) = (1/2) * N₀
N(t) = N₀ * e^(-kt)
(1/2) * N₀ = N₀ * e^(-k * 42)
Canceling out N₀ from both sides, we get:
1/2 = e^(-k * 42)
Taking the natural logarithm of both sides, we have:
ln(1/2) = -k * 42
Solving for k, we find:
k = -ln(1/2)/42
Now we can plug in the values of N₀, t, and k to find N(t). We want to find the remaining amount of material after 8 days, so t = 8.
N(8) = N₀ * e^(-k * 8)
N(8) = 49 * e^(-(-ln(1/2)/42) * 8)
Evaluating this expression, we find that approximately 44.063 kg of radioactive material remains after 8 days.