The half-life of caffeine in the human body is about 5.7 hours. A cup of coffee has about $90 \mathrm{mg}$ of caffeine.
a. Write an equation for the amount of caffeine in a person's body after drinking a cup of coffee? Let $C$ be the milligrams of caffeine in the body after $t$ hours.
b. How much caffeine will remain after 10 hours?
$\mathrm{mg}$
State your answer to the nearest hundredth of a $\mathrm{mg}$.
c. How long until there are only $20 \mathrm{mg}$ remaining?
hours
State your answer to the nearest hundredth of an hour.

Step 1 ：The amount of caffeine in a person's body after drinking a cup of coffee can be modeled by an exponential decay function. The general form of an exponential decay function is \(y = a(0.5)^{t/h}\), where \(a\) is the initial amount, \(t\) is the time, and \(h\) is the half-life. In this case, the initial amount of caffeine is \(90 \mathrm{mg}\), and the half-life is \(5.7\) hours. So, the equation for the amount of caffeine in a person's body after drinking a cup of coffee is \(C = 90(0.5)^{t/5.7}\).

Step 2 ：To find out how much caffeine will remain after 10 hours, we can substitute \(t = 10\) into the equation.

Step 3 ：\(a = 90\)

Step 4 ：\(h = 5.7\)

Step 5 ：\(t = 10\)

Step 6 ：\(C = 26.68\)

Step 7 ：\(C = 26.68\)

Step 8 ：Final Answer: The amount of caffeine remaining in the body after 10 hours is approximately \(\boxed{26.68}\) mg.