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 $32.3 mg$ 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.
$y = ◻^{D t}$

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$y = 32.3 \cdot 2^{\frac{t}{15}}$ is the final answer.

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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 $32.3 mg$ 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 ： $y = 32.3 \cdot 2^{\frac{t}{15}}$ is the final answer.

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