Answer to Question #270826 in Differential Equations for dan

Question

Gerry is to memorize 150 terms for a quiz. If the rate by which the number of terms Gerry memorizes is directly proportional to the number of terms already memorized and the number of remaining terms; and that initially Gerry knows 5 terms and 20 terms 5 minutes later. At what time is he half done?

Answer 1

$\displaystyle \text{Let m be the number terms Gerry memorizes, therefore}\\ \frac{dm}{dt}=km\\ \implies \frac{dm}{m}=kdt\\ \text{Integrating, we have that}\\ \ln m = kt + c\\ \text{Taking the exponential of both sides}\\ m = Ae^{kt}\\ \text{At t = 0, m =5}\\ A= 5\\ \implies m = 5 e^{kt}\\ \text{At t = 5, m =20}\\ m = 5e^{5k}\\ e^{5k}=4\\ \implies k = \frac{\ln 4}{5} = 0.277\\ 75 = 5e^{0.277t}\\ \implies t = 9.776 \approx 10 min$