As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, $a$.

A. Let $y=f(t)$ be the fraction of the original material remembered $t$ weeks after the course has ended. Set up a differential equation for $y$, using $k$ as any constant of proportionality you may need (let $k>0$). Your equation will contain two constants; the constant $a$ (also positive) is less than $y$ for all $t$.
$\frac{dy}{dt} =$
What is the initial condition for your equation?
$y(0) =$

B. Solve the differential equation.
$y =$

C. What are the practical meaning (in terms of the amount remembered) of the constants in the solution $y=f(t)$? If after one week the student remembers 88 percent of the material learned in the semester, and after two weeks remembers 81 percent, how much will she or he remember after summer vacation (about 14 weeks)?
percent =