Question

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.

Answer 1

Answer:

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