$(1$ point$)$ According to a simple physiological model, an athletic adult male needs $20$ calories per day per pound of body weight to maintain his weight. If he consumes more or fewer calories than those required to maintain his weight, his weight changes at a rate proportional to the difference between the number of calories consumed and the number needed to maintain his current weight; the constant of proportionality is $1/35001/3500$ pounds per calorie. Suppose that a particular person has a constant caloric intake of HH calories per day. Let W$($t$)$W$($t$)$ be the person's weight in pounds at time tt $($measured in days$).$

$($a$)$ What differential equation has solution W$($t$)$W$($t$)?$

dWdt$=$dWdt$=$

$($Your answer may involve W$,$ H and values given in the problem.$)$

$($b$)$ Solve this differential equation, if the person starts out weighing $180$ pounds and consumes $3800$ calories a day.

W$=$W$=$

$($c$)$ What happens to the person's weight as t$$t$$$?$

W$$W$$