Exponenting and Logurithmic functions
Writing and evaluating a function modeling continuous exponential growt.
$0B$
A sample of a radioactive substance has an initial mass of $78.4\mathrm{mg}$. This substance follows a continuous exponential decay model and has a half-life of 15 minutes.
(a) Let $f$ be the time (in minutes) since the start of the experiment, and let $y$ be the amount of the substance at time $t$.
Write a formula relating y to fi
Use exact expressions to fill in the missing parts of the formula. Do not use approximations.
$$y=78+e^{(15)}$$
(b) How much will be present in 23 minutes?
Do not round any intermediate computations, and round your answer to the nearest tenth.

Step 1 ：The problem provides that a sample of a radioactive substance has an initial mass of 78.4 mg, follows a continuous exponential decay model, and has a half-life of 15 minutes. We are asked to find a formula relating the amount of the substance y to the time t, and to calculate how much of the substance will be present in 23 minutes.

Step 2 ：First, we can express the continuous exponential decay mathematically using the formula $y=a\cdot (1/2{)}^{t/h}$, where $a$ is the initial amount, $t$ is the time, and $h$ is the half-life.

Step 3 ：Substituting the given values into the formula, we get $y=78.4\cdot (1/2{)}^{t/15}$. This formula represents the amount of the substance at any given time $t$.

Step 4 ：To find out how much of the substance will be present in 23 minutes, we substitute $t=23$ into the formula, which gives $y=78.4\cdot (1/2{)}^{23/15}$.

Step 5 ：Calculating the above expression, we find that the amount of the substance present in 23 minutes will be approximately 27.1 mg.

Step 6 ：So, the formula relating $y$ to $t$ is $y=78.4\cdot (1/2{)}^{t/15}$, and the amount of the substance present in 23 minutes will be approximately 27.1 mg.