Here is the entire problem below:
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 1/3500 pounds per calorie. Suppose that a particular person has a constant caloric intake of H calories per day. Let W(t) be the person's weight in pounds at time t (measured in days).
(a) What differential equation has solution W(t)?
So I already solved for this and got that part correct. My answer was: $$\frac{dW}{dt} = \frac{1}{3500}(H-20W)$$
Below is the part that I am unsure of what to do:
(b) If the person starts out weighing 165 pounds and consumes 3500 calories a day,$$\lim_{t\to \infty} W(t) = \text{?}$$
I at first thought to plug in the H(3500) and W(165) values that were given in the part b question into the answer I got for part A. But then I wasn't sure what to do with that answer, or if that is even the way to go. It's probably a simple solution, but I'm not seeing where to go from here. Any help would be appreciated. Thank you.
Best Answer
When the weight stabilizes (at infinity), the derivative cancels out and
$$H-20W_\infty=0.$$