Jane is really out of shape and currently runs at a 3 mile per hour pace. Jane begins training on Jan. 1, 2013; her pace improves dramatically, although she will never be able to run faster than 7 miles per hour. Let v(t) be Janes pace (in miles per hour) t days after Jan. 1.

a) Assuming that her pace v(t) grows according to a logistic model, write down an IVP for v(t). Your DE will include the constant k, as you do not yet know its value.

b) After 100 days of training, Jane runs a 5 mile/hr pace. Find an explicit formula for v(t), then plot it. How long will it take for Jane to have achieve a 6 mile/hr pace? (Hint: you will have to calculate k...)