$(1$ point$)$ 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, $\backslash ($ a $\backslash ).$

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

$\backslash [$

$\backslash$frac$\{$d y$\}\{$d t$\}=$

$\backslash ]$

What is the initial condition for your equation?

$\backslash [$

y$(0)=$

$\backslash ]$

B$.$ Solve the differential equation.

$\backslash [$

y$=$

$\backslash ]$

C$.$ What are the practical meaning $($in terms of the amount remembered$)$ of the constants in the solution $\backslash ($ y$=$f$($t$)\backslash )?$ If after one week the student remembers $90$ percent of the material learned in the semester, and after two weeks remembers $84$ percent, how much will she or he remember after summer vacation $($about $14$ weeks$)?$

percent $=$