A certain substance decomposes according to a continuous exponential decay model. To begin an experiment, the initial amount of the substance is $400 \mathrm{~kg}$. After 12 hours, $260 \mathrm{~kg}$ of the substance is left.
(a) Let $t$ be the time (in hours) since the beginning of the experiment, and let $y$ be the amount of the substance (in $\mathrm{kg}$ ) 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.

Step 1 ：The problem describes a situation of exponential decay. The general formula for exponential decay is given by: \(y = a \cdot e^{kt}\) where: \(y\) is the amount of substance at time \(t\), \(a\) is the initial amount of the substance, \(k\) is the decay constant, and \(t\) is the time since the beginning of the experiment.

Step 2 ：We know that the initial amount of the substance (\(a\)) is 400 kg, and after 12 hours (\(t\)), 260 kg of the substance is left (\(y\)). We can use these values to solve for the decay constant \(k\).

Step 3 ：Let's calculate the decay constant \(k\) first. Using the values \(a = 400\), \(y = 260\), and \(t = 12\), we find that \(k = -0.03589857634103785\).

Step 4 ：Now that we have the decay constant \(k\), we can substitute it back into the general formula to get the specific formula relating \(y\) to \(t\) for this substance. Using the values \(a = 400\), \(k = -0.03589857634103785\), we get the formula \(y(t) = 400 \cdot e^{-0.0359t}\).

Step 5 ：\(\boxed{y(t) = 400 \cdot e^{-0.0359t}}\) is the final answer. This formula can be used to calculate the amount of the substance left at any time \(t\) since the beginning of the experiment.