Questions: The mass of a radioactive substance follows a continuous exponential decay model. A sample of this radioactive substance has an initial mass of 1989 log and decreases continuously at a relative rate of 19% 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 1989 log and decreases continuously at a relative rate of 19% 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 1989 log and decreases continuously at a relative rate of $19 \%$ 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.

Solution

Solution Steps

Step 1: Understand the Problem

We are given the initial mass of a radioactive substance, its continuous exponential decay rate per day as a percentage, and the time in days after which we want to find the remaining mass. We need to apply the formula for continuous exponential decay.

Step 2: Apply the Continuous Exponential Decay Formula

The formula to calculate the mass of a radioactive substance after a certain time period given its initial mass and decay rate is: $$ M(t) = M_0 \cdot e^{(-r/100) \cdot t} $$ where:

$M(t)$ is the mass of the substance after $t$ days,
$M_0$ is the initial mass of the substance,
$r$ is the decay rate per day as a percentage,
$t$ is the time in days,
$e$ is the base of the natural logarithm.

Step 3: Substitute the Given Values and Calculate

Given that $M_0 = 1989$, $r = 19\%$, and $t = 6$ days, we substitute these values into the formula: $$ M(6) = 1989 \cdot e^{(-19/100) \cdot 6} $$ After calculation, the mass of the radioactive substance after 6 days is approximately 636.1 units.