The amount of caffeine in the bloodstream decreases by 27 ev

The amount of caffeine in the bloodstream decreases by 27% every hour. a. A cup of coffee contains 94 mg of caffeine. Define a function, f that gives the amount of caffeine remaining in the blood stream t hours after the coffee was consumed. b. How many hours will it take for the amount of caffeine to reach half of the amount in the initial cup of coffee? hours c. A double shot of espresso contains 128 mg of caffeine. Define a function, g, that gives the amount of caffeine remaining in the blood stream x hours after the espresso was consumed. d. How many hours will it take for the amount of caffeine to reach half of the amount in the initial espresso? hours. e. A can of Rock Star energy drink has 170 mg of caffeine in it. How long after you consume the Rock Star will the caffeine in your blood stream decrease to 85 mg? hours f. True or False: The number of hours it takes for the amount of caffeine in your body to reach half of the amount initially in your body depends on the initial amount of caffeine in your body.

Solution

amount decreases by 27% every hour => r=27_/100 =0.27

a)initial amount =94 mg

amount of caffine remaining in blood stream,f(t)=94*(1-0.27)^t

amount of caffine remaining in blood stream,f(t)=94*(0.73)^t

b) half of initial amount =94_/2=47mg

47=94*(0.73)^t

(0.73)^t=(47_/94)

(0.73)^t=0.5

t=ln_0.730.5

t=2.2hours

c)

initial amount =128 mg

amount of caffine remaining in blood stream,g(x)=128*(1-0.27)^x

amount of caffine remaining in blood stream,g(x)=128*(0.73)^x

d)

half of initial amount =128_/2=64mg

64=128*(0.73)^t

(0.73)^t=(64_/128)

(0.73)^t=0.5

t=ln_0.730.5

t=2.2hours

e)

initial amount =170mg

85=170*(0.73)^t

(0.73)^t=(85_/170)

(0.73)^t=0.5

t=ln_0.730.5

t=2.2hours

f)

False, instead it depends on rate of decrease