The mass of a radioactive substance follows an exponential decay model, with a decay rate of 5% per day. A sample of this radioactive substance was taken six days ago. If the sample has a mass of 3 kg today, find the initial mass of the sample. Round your answer to two decimal places.
To find the initial mass of the sample, we can use the exponential decay model. The formula for exponential decay is:
M(t) = M₀ * e^(r * t),
where M(t) is the mass at time t, M₀ is the initial mass, r is the decay rate, and t is the time elapsed.
In this case, we know the current mass M(t) is 3 kg, the decay rate r is 5% per day (which can be written as 0.05), and the time elapsed t is 6 days. We need to solve for the initial mass M₀.
The formula becomes:
3 = M₀ * e^(0.05 * 6).
To isolate M₀, we divide both sides of the equation by e^(0.05 * 6):
3 / e^(0.05 * 6) = M₀.
Using a calculator, evaluate e^(0.05 * 6) to get approximately 1.340482543. Divide 3 by that value:
3 / 1.340482543 ≈ 2.237.
Therefore, the initial mass of the sample is approximately 2.24 kg (rounded to two decimal places).
3 = a(.95)^6
a = 3/.95^6 = 4.08 kg