Writing and evaluating a function modeling continuous exponential growt.
A sample of a radioactive substance has an initial mass of $151.4\mathrm{mg}$. This substance follows a continuous exponential decay model and has a half-life of 5 days.
(a) Let $t$ be the time (in days) since the start of the experiment, 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.
$$y=◻e^{\mathbb{(}\mathbb{1}\mathbb{D}}t$$
(b) How much will be present in 12 days?
Do not round any intermediate computations, and round your answer to the nearest tenth.
○mg
Explanation
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Step 1 ：The formula for exponential decay is given by: $y=y_0\ast e^{kt}$ where $y$ is the final amount of the substance, $y_0$ is the initial amount of the substance, $e$ is the base of the natural logarithm, $k$ is the decay constant, and $t$ is the time.

Step 2 ：The decay constant $k$ can be calculated using the half-life of the substance. The formula for the decay constant is: $k=\frac{ln(2)}{T}$ where $T$ is the half-life of the substance.

Step 3 ：In this case, the initial amount of the substance $y_0$ is 151.4 mg, the half-life $T$ is 5 days, and we want to find the amount of the substance after 12 days.

Step 4 ：Substituting the given values into the formula for the decay constant, we get $k=\frac{ln(2)}{5}=0.139$

Step 5 ：Substituting the values of $y_0$, $k$, and $t$ into the formula for exponential decay, we get $y=151.4\ast e^{-0.139\ast 12}=28.7$ mg

Step 6 ：Final Answer: The formula relating $y$ to $t$ is $y=151.4\ast e^{-0.139t}$. After 12 days, there will be approximately $\boxed{{\displaystyle 28.7\text{ mg}}}$ of the substance left.