Questions: The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 167 kg and decreases continuously at a relative rate of 8% per day. Find the mass of the sample after six days. Do not round any intermediate computations, and round your answer to the nearest tenth.

The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 167 kg and decreases continuously at a relative rate of 8% per day. Find the mass of the sample after six days.
Do not round any intermediate computations, and round your answer to the nearest tenth.

Transcript text: The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 167 kg and decreases continuously at a relative rate of $8 \%$ per day. Find the mass of the sample after six days. Do not round any intermediate computations, and round your answer to the nearest tenth. $\square$ D $k$

Solution

Solution Steps

Step 1: Convert the Relative Decay Rate from Percentage to Decimal

The given relative decay rate is 8%. To use it in calculations, we convert it to a decimal by dividing by 100. \[r = \frac{8}{100} = 0.08\]

Step 2: Use the Continuous Exponential Decay Formula

The formula to calculate the final mass of a radioactive substance after a certain period is given by: \[M = M_0 \cdot e^{(-r \cdot t)}\] where $M_0$ is the initial mass (167 units), $r$ is the decay rate per time unit as a decimal (0.08), and $t$ is the time elapsed (6 units). Substituting the given values into the formula, we get: \[M = 167 \cdot e^{(-0.08 \cdot 6)}\] \[M = 103.3\] after rounding to 1 decimal places.