The problem
I need help with the following.. math word problem. I really appreciated your help!
The half-life of caffeine in a typical person is 3 hours. José consumes 200 mg at 8 am, another 200 mg at 10 am, and another 200 mg at 4 pm. How much caffeine will José still have in his system at bedtime (10 pm)?
Answer provided by our tutors
The amount A of caffeine present at a time t is given by the model:
A = A0*(1/2)^(t/h)
where A0 is the amount present initially, h is the half-life of the material.
In our case
A0 = 200 mg
h = 3 hr
The amount of caffein left at 10 am is:
A = 200(1/2)(t/3)
For 10 am we have t = 2 hr and we plug that value in the previous equation:
A = 200(1/2)(2/3)
A = 200/3 mg
Since Jose cosume another 200 mg now he has total of: 200/3 + 200 mg caffeine
At 4 pm, that is 6 hours ater 10 am we have:
t = 6 hr
A0 = (200/3 + 200) mg
A = (200/3 + 200)(1/2)(6/3)
A = 800/3 mg
Since Jose cosume another 200 mg, at 4 pm he has 800/3 + 200 mg affeine.
At 10 pm, that is 6 hours after 4 pm we have:
t = 6 hr
A0 = 800/3 + 200
A = (800/3 + 200)(1/2)(6/3)
A = 1400/3 mg
A = 466.67 mg
At bed time. 10 pm, Jose will still have 466.67 mg caffeine.