Exponenting and Logurithmic functions
Writing and evaluating a function modeling continuous exponential growt.
$0 B$
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.