Question

The mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of 3% per day. A sample of this radioactive substance was taken three days ago. If the sample has a mass of 4 kg today, find the initial mass of the sample. Round your answer to two decimal places. Note: This is a continuous exponential decay model. And though the decay rate parameter is 3% per day, the actual decay is not 3% each day. kg Х 5 ?

Answer 1

The initial mass of the radioactive sample was approximately 4.37 kg, calculated using the exponential decay formula with a 3% decay rate parameter per day and the known current mass of 4 kg after 3 days.

Step-by-step explanation:

The mass of a radioactive substance decreasing over time can be described using an exponential decay model. In this particular case, the decay rate parameter is 3% per day. To find the initial mass of the sample, we use the formula for exponential decay: m(t) = m0e-kt

Where:

• m(t) is the mass after time t
• m0 is the initial mass
• k is the decay rate parameter (0.03 per day in this case)
• t is the time (in days) that has passed

Given that the sample has a mass of 4 kg today (m(t) = 4 kg) and that it has been 3 days since the sample was taken, we can solve for m0 as follows:

1. Substitute the known values into the decay formula:
2. 4 kg = m0e-0.03 × 3
3. Solve for m0:
4. m0 = 4 kg / e-0.09
5. Calculate using the exponential function:
6. m0 = 4 kg / 0.9139 (approx)
7. Round to two decimal places:
8. m0 = 4.374 kg (round to 4.37 kg)

Therefore, the initial mass of the radioactive sample was approximately 4.37 kg.