An adult consumes an espresso containing 60 mg of caffeine. If the caffeine has a half-life of 5.5 hours, when will the amount of caffeine in her bloodstream equal 30 mg? Create an exponential model for the given situation. Use the exact value for k in the function.

since the caffeine concentration drops to half, the time required is the half-life = 5.5 hours

Step-by-step explanation:

if we use an exponential model

C (t) = C (0) * e^ (-kt)

where

C (t) = concentration at time t

C (0) = concentration at time t=0

k = characteristic parameter

then, when the caffeine concentration reaches half-life (t=th), the concentration will be half of the initial, therefore

C (t) = C (0) / 2 = C (0) * e^ (-kT)

- ln 2 = - th*k

k = ln (2) / th

then when the concentration reaches C₁=30 mg

C₁ = C (0) * e^ (-kt)

t = ln [C (0) / C₁] / k = [ln [C (0) / C₁] / ln (2) ] * th

replacing values

t = [ln [C (0) / C₁] / ln (2) ] * th = [ln [60 mg / 30 mg] / ln (2) ] * 5.5 hours = 5.5 hours

since the caffeine concentration drops to half, the time required is the half-life = 5.5