The half-life caffeine in the human body is typically about 6 hours. If john drinks a large cafe mocha with an extra shot of expresso from a coffee shop at 8am (255mg caffeine), and then starts to feel drowsy around 4pm and drinks another large cafe mocha (no expresso, 180mg caffeine). Can anybody help me how to do this step by step? PLEASE
a)how much caffeine is in johns bloodstream at 11:30 am?
b) How much caffeine is in Mary's bloodstream at 4:30 pm?
amount = intitial amount (1/2)^(t/6), where t is hours and initial amount = 255
from 8:00 to 11:30 = 3.5 hours
from 8:00 to 4:00 pm = 8 hours
a) at 11:30, t = 3.5
amount of caffeine = 255(1/2)^(3.5/6) =170.192
b) at 4:00
amount = 255(.5)^(8/6) = 101.1968
she drinks another coffee containing 180 mg
so now she has 101.1968 + 180 or 281.1968 mg in her
half our later ,
amount left = 281.1968(.5)^(1/12) = appr 265.4 mg
To calculate the amount of caffeine in John's bloodstream at a specific time, we need to consider the half-life of caffeine and the amount consumed at each time.
a) To find out how much caffeine is in John's bloodstream at 11:30 am, we need to determine how many half-lives have passed since he consumed the coffee.
Step 1: Determine the number of hours that have passed since John consumed the coffee. The time between consumption (8 am) and 11:30 am is 3.5 hours.
Step 2: Divide the number of hours passed by the half-life of caffeine (6 hours) to find the number of half-lives. In this case, 3.5 / 6 = 0.5833.
Step 3: Calculate the amount of caffeine remaining in John's body after each half-life. Since the initial amount consumed was 255mg, after 0.5833 half-lives, the caffeine remaining can be found using the formula:
Remaining Caffeine = Initial Caffeine * (1/2)^(Number of Half-Lives)
In this case, Remaining Caffeine = 255mg * (1/2)^0.5833
Using a scientific calculator, we can calculate that the remaining caffeine at 11:30 am is approximately 216.37 mg.
Therefore, the approximate amount of caffeine in John's bloodstream at 11:30 am is 216.37 mg.
b) To calculate the amount of caffeine in Mary's bloodstream at 4:30 pm, we need to consider her consumption of the second cafe mocha.
Step 1: Determine the number of hours that have passed since Mary consumed the second cafe mocha. The time between consumption (4 pm) and 4:30 pm is 0.5 hours.
Step 2: Divide the number of hours passed by the half-life of caffeine (6 hours) to find the number of half-lives. In this case, 0.5 / 6 = 0.0833.
Step 3: Calculate the amount of caffeine remaining in Mary's body after each half-life. Since the initial amount consumed was 180mg, after 0.0833 half-lives, the caffeine remaining can be found using the formula mentioned above.
Remaining Caffeine = 180mg * (1/2)^0.0833
Using a scientific calculator, we can calculate that the remaining caffeine at 4:30 pm is approximately 179.81 mg.
Therefore, the approximate amount of caffeine in Mary's bloodstream at 4:30 pm is 179.81 mg.