The mass of a substance, which follows a continuous exponential growth model, is being studied in a lab. The doubling time for this substance was observed to be 15 hours. There were $32.3 \mathrm{mg}$ of the substance present at the beginning of the study.
(a) Let $t$ be the time (in hours) since the beginning of the study, and let $y$ be the amount of the substance at time $t$.
Write a formula relating $y$ to $t$. Use exact expressions to fill in the missing parts of the formula. Do not use approximations.
\[
y=\square^{\mathbb{D} t}
\]
Answer
\(\boxed{y = 32.3 \cdot 2^{\frac{t}{15}}}\) is the final answer.
Step 1 ：Given that the substance follows a continuous exponential growth model and its doubling time is 15 hours, we can use the formula for exponential growth, which is \(y = a \cdot b^{t}\), where \(a\) is the initial amount of the substance and \(b\) is the growth factor.
Step 2 ：In this case, the initial amount \(a\) is \(32.3 \mathrm{mg}\) and the growth factor \(b\) can be calculated as \(2^{1/15}\), because the substance doubles every 15 hours.
Step 3 ：So, the formula relating \(y\) to \(t\) is \(y = 32.3 \cdot 2^{t/15}\).
Step 4 ：\(\boxed{y = 32.3 \cdot 2^{\frac{t}{15}}}\) is the final answer.
