Questions: The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 2542 kg and decreases continuously at a relative rate of 13% per day. Find the mass of the sample after four days. Do not round any intermediate computations, and round your answer to the nearest tenth.

Solution Steps

The mass of a radioactive substance that decays exponentially can be modeled by the equation:

\[ m(t) = m_0 \cdot e^{-kt} \]

where:

• \( m(t) \) is the mass at time \( t \),
• \( m_0 \) is the initial mass,
• \( k \) is the decay constant,
• \( t \) is the time in days.

From the problem, we have:

• Initial mass, \( m_0 = 2542 \) kg,
• Relative decay rate, \( 13\% \) per day, which means \( k = 0.13 \),
• Time, \( t = 4 \) days.

Substitute the given values into the formula:

\[ m(4) = 2542 \cdot e^{-0.13 \times 4} \]

Calculate the exponent:

\[ -0.13 \times 4 = -0.52 \]

Now calculate \( e^{-0.52} \):

\[ e^{-0.52} \approx 0.5945 \]

Substitute the value of \( e^{-0.52} \) back into the equation:

\[ m(4) = 2542 \cdot 0.5945 \approx 1510.459 \]

Round the mass to the nearest tenth:

\[ m(4) \approx 1510.5 \text{ kg} \]

Final Answer