A certain drug is eliminated from the bloodstream with a half-life of 32 hours. Suppose that a patient receives an initial dose of $60 \mathrm{mg}$ of the drug at midnight. a. How much of the drug is in the patient's blood at noon later that day? b. When will the drug concentration reach $35 \%$ of its initial level? a. The reference point is midnight. If $\mathrm{t}$ is measured in hours, what is the exponential decay function? \[ y(t)= \] (Type an expression. Do not round until the final answer. Then round coefficients to six decimal places as needed.)

Step 1 ：The half-life of a substance undergoing exponential decay can be used to determine the decay constant, which is the coefficient in the exponential term of the decay function. The decay function has the form \(y(t) = y0 * e^{kt}\), where \(y0\) is the initial amount of the substance, \(t\) is time, \(k\) is the decay constant, and \(e\) is the base of the natural logarithm.

Step 2 ：The decay constant can be found using the formula \(k = \ln(2) / \text{half-life}\). In this case, the half-life is 32 hours, so the decay constant is \(\ln(2) / 32\).

Step 3 ：The initial amount of the drug is 60 mg, so the decay function is \(y(t) = 60 * e^{(\ln(2) / 32) * t}\).

Step 4 ：To find out how much of the drug is in the patient's blood at noon later that day, we need to substitute \(t = 12\) into the decay function.

Step 5 ：The remaining dose of the drug in the patient's blood at noon, 12 hours after the initial dose was administered, is approximately 46.27 mg.

Step 6 ：Final Answer: The amount of the drug in the patient's blood at noon later that day is approximately \(\boxed{46.27 \, \text{mg}}\).