Suppose you meet 100 new people each year, but each year you forget 40 percent o

Question

Suppose you meet 100 new people each year, but each year you forget 40 percent of all the people that you know. Create and solve the differential equation that explains the rate of change of the number of people P that you know at any time t. Solve the differential equation subject to the initial condition that you didn't know anyone at the time of your birth. Given your age, and according to the model, how many people do you know?

Explanation / Answer

rate of increase of knowing persons = 100 per year rate of decrease of knowing persons = 0.4P per year where P is the number of persons known therefore net rate of increase in knowing of persons, dP/dt = 100 - 0.4P dP/100-0.4P = dt 100 - 0.4P = X -0.4dP = dX dP = -dX/0.4 -dX/0.4X = dt dX/X = -0.4dt ln X = -0.4t + c at t = 0, P = 0 => X = 100 ln 100 = c => ln (X/100) = -0.4t => X/100 = e^(-0.4t) => (100-0.4P)/100 = e^(-0.4t) => 1-0.004P = e^(-0.4t) =>1-e^(-0.4t) = 0.004P =>250(1-e^(-0.4t)) = P

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