Math questions! Help ASAP

Question

Part A:

Reshanda was invited for coffee at 2:00 pm, but she normally goes to sleep at 10:00 pm. If a 16-oz coffee has 122 mg of caffiene, and the half-life of caffiene is 5.5 hours, how much caffiene would still be in her system at bedtime? (Round your answer to the nearest tenth of a milligram)

Part B:

Suppose Reshanda cannot sleep if she has 15 or more mg of caffeine in her system. How many hours before she goes to sleep could she drink a 16-oz coffee? (Round your answer to the nearest tenth of an hour)

Answer 1

half life = 5.5

1/2 = e^5.5r

ln.5 = 5.5r

r = ln.5/5.5 = about -12.6% decay rate hourly

122 would decay in 8 hours to

A = 122e^-.126(8)= 122/e^1.008

=about 44.52 mg of caffeine at 10:00 pm

15 = 122e^-.126h where h=hours to reach 15 mg caffeine

15/122 = e^-.126h

-.126h = ln(15/122)

h = ln(15/122)/-.126 = about 16.6 hours

she won't get to sleep until 2pm +16.6 hours =

about 6:36 am

using the half life at 5.5 hours

2pm 122

add 5.5 hours and divide 122 by 2

7:30pm 61

again add 5.5 and divide by 2

1am 30.5

one more half life gets to

6:30 am 15.25 mg

slightly past 6:30 am caffeine will reach 15 mg

6:36 am looks about right