Respuesta :
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:
[tex]N(t)=N_0e^{-\lambda t}[/tex]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:
[tex]\begin{gathered} \frac{N_0}{2}=N_0e^{-\lambda\cdot2} \\ \frac{1}{2}=e^{-2\lambda} \\ \ln (\frac{1}{2})=-2\lambda\ln (e)=-2\lambda_{} \\ \lambda=\frac{\ln (2)}{2} \end{gathered}[/tex]In our case, the initial amount of chocolate is 75 grams and the amount of chocolate after 7 hours is:
[tex]N=75\cdot e^{-\frac{\ln(2)}{2}\cdot7}\approx6.63[/tex]So, at 11 pm yo will have 6.63 grams of chocolate and will have a problems to falling sleep.
