Question

Pregnant women metabolize some drugs at a slower rate than the rest of the population. The half-life of caffeine is about 4 hours for most people. In pregnant women, it is 10 hours1. (This is important because caffeine, like all psychoactive drugs, crosses the placenta to the fetus.) If a pregnant woman and her husband each have a cup of coffee containing 110 mg of caffeine at 8 am, how much caffeine does each have left in the body at 7 pm?

Answer 1

Husband:

The husband will have 16.35 mg of caffeine in his body at 7 pm.

Woman:

The pregnant woman will have 51.33 mg of caffeine in her body at 7 pm.

Explanation:

The amount of caffeine in the body can be modeled by the following equation:

`C(t) = C(0)e^(rt)`

In which C(t) is the amount of caffeine t hours after 8 am, C(0) is how much coffee they took and r is the rate the the amount of caffeine decreases in their bodies.

110 mg of caffeine at 8 am,

So
`C(0) = 110`

Husband

Half life of 4 hours. So

`C(4) = 0.5C(0) = 0.5*110 = 55`

`C(t) = C(0)e^(rt)`

`55 = 110e^(4r)`

`e^(4r) = 0.5`

Applying ln to both sides

`\ln{e^(4r)} = ln(0.5)`

`4r = ln(0.5)`

`r = (ln(0.5))/(4)`

`r = -0.1733`

So for the husband

`C(t) = 110e^(-0.1733t)`

At 7 pm

7 pm is 11 hours after 8 am, so this is C(11)

`C(t) = 110e^(-0.1733t)`

`C(11) = 110e^(-0.1733*11) = 16.35`

The husband will have 16.35 mg of caffeine in his body at 7 pm.

Pregnant woman

Half life of 10 hours. So

`C(10) = 0.5C(0) = 0.5*110 = 55`

`C(t) = C(0)e^(rt)`

`55 = 110e^(10)`

`e^(10r) = 0.5`

Applying ln to both sides

`\ln{e^(10r)} = ln(0.5)`

`10r = ln(0.5)`

`r = (ln(0.5))/(10)`

`r = -0.0693`

At 7 pm

7 pm is 11 hours after 8 am, so this is C(11)

`C(t) = 110e^(-0.0693t)`

`C(11) = 110e^(-0.0693*11) = 51.33`

The pregnant woman will have 51.33 mg of caffeine in her body at 7 pm.