Final answer:
The equation for the amount of caffeine remaining in a person's body after t hours of consuming a cup with 115mg of caffeine is A(t) = 115(1/2)^(t/5.5), where A(t) is the caffeine amount in milligrams and t is time in hours.
Step-by-step explanation:
To derive the equation for the amount of caffeine in a person's body after drinking a cup containing 115mg of caffeine, we can use the concept of exponential decay, which is applicable because caffeine has a half-life of about 5.5 hours. The formula for exponential decay is A(t) = A_0(1/2)^(t/T), where A(t) is the amount of substance at time t, A_0 is the initial amount, (1/2) is the decay factor for each half-life, and T is the half-life period.
For caffeine, if we take A_0 as 115mg and T as 5.5 hours, the formula becomes A(t) = 115(1/2)^(t/5.5). Here, A(t) will give us the remaining amount of caffeine in milligrams in the body after t hours since consuming the coffee.