Question

the mass of a radioactive substance follows a continuous exponential decay model, with a decay rate parameter of 4% per day. a sample of this radioactive substance was taken two 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 4% per day, the actual decay is not 4% each day.

Explanation:

Step1: Recall decay formula

The continuous exponential decay model is \( m(t) = m_0 e^{-rt} \), where \( m(t) \) is the mass at time \( t \), \( m_0 \) is the initial mass, \( r \) is the decay rate, and \( t \) is time.

Step2: Identify known values

We know \( m(t) = 4 \) kg (mass today), \( r = 0.04 \) (4% decay rate), and \( t = 2 \) days (time since initial sample). We need to find \( m_0 \).

Step3: Rearrange formula for \( m_0 \)

From \( m(t) = m_0 e^{-rt} \), solve for \( m_0 \): \( m_0 = \frac{m(t)}{e^{-rt}} = m(t) e^{rt} \).

Step4: Substitute values

Substitute \( m(t) = 4 \), \( r = 0.04 \), and \( t = 2 \) into the formula: \( m_0 = 4 \times e^{0.04 \times 2} \).

Step5: Calculate exponent and \( e \)-value

First, calculate the exponent: \( 0.04 \times 2 = 0.08 \). Then, \( e^{0.08} \approx 1.083287 \).

Step6: Find \( m_0 \)

Multiply by 4: \( m_0 = 4 \times 1.083287 \approx 4.333148 \). Round to two decimal places.

Answer:

\( 4.33 \) (kg)