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.
\[
y=\square e^{\mathbb{( 1 D}} t
\]
(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: \(y = y_0 * e^{kt}\) 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 \(y = 151.4 * e^{-0.139*12} = 28.7\) mg

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