Question

Chocolate is considered a stimulant and make make it difficult to fall asleep if you have 6 grams of it in your system at bedtime. The half life of chocolate is 2 hours. If you ate a chocolate bar that weighed 75 grams at 4pm, would you hav a difficult time falling asleep at 11pm? Make sure to mathematically justify your answer.

Answer 1

This is an exponential decay problem. We have that the half life of chocolate is 2 hours, this means that after two hours the amount of chocolate reduce by a factor of 2.

The formula for an exponential decay is:

`N(t)=N_0e^(-\lambda t)`

Where N(t) is the amount of chocolate at time t, N0 is the initial amount of chocolate, t is time and lamda is a constant. We can find the constant lamda with the half life time of 2 hours, so:

`\begin{gathered} (N_0)/(2)=N_0e^(-\lambda\cdot2) \\ (1)/(2)=e^(-2\lambda) \\ \ln ((1)/(2))=-2\lambda\ln (e)=-2\lambda_{} \\ \lambda=(\ln (2))/(2) \end{gathered}`

In our case, the initial amount of chocolate is 75 grams and the amount of chocolate after 7 hours is:

`N=75\cdot e^{-(\ln(2))/(2)\cdot7}\approx6.63`

So, at 11 pm yo will have 6.63 grams of chocolate and will have a problems to falling sleep.