Mathematics

Problem

A researcher measures the rate at which a chemical diffuses through a membrane. The rate (in mg/s) at time

t

seconds is given by

R(t)=\frac{e^{3t}}{1+e^{3t}}, \qquad 0\le t\le \ln 4.

(a) Determine the total mass of chemical that passes through the membrane during the time interval

0\le t\le \ln 4

. (b) Interpret the numerical result of part (a) in the context of the experiment, including appropriate units. (a) Evaluate

\displaystyle \int_{0}^{\ln 4}\!R(t)\,dt

exactly. (b) State the total mass that has diffused through the membrane and give its units.