A social media rumor is spreading through a university campus of 2500 students. Initially, 50 students have heard the rumor. The number of students who have heard the rumor is observed to increase by 40% each hour.
The total number of students who have heard the rumor, xxx, is modelled by
x=A×Bt x = A \times B^t x=A×Btwhere A A\,A and B B\,B are constants and t t\,t is the number of hours after the rumor first started.
(i) Based on this model, calculate the number of students who have heard the rumor 5 hours after it started.
(ii) Explain why this model is not appropriate for predicting the spread of the rumor over a long period of time.
A more sophisticated model assumes that the rate at which the rumor spreads is given by the differential equation
dxdt=x(2500−x)5000 \frac{dx}{dt} = \frac{x(2500 - x)}{5000} dtdx=5000x(2500−x)(i) Show that
∫(Cx+D2500−x)dx=∫dt \int \left( \frac{C}{x} + \frac{D}{2500 - x} \right) dx = \int dt ∫(xC+2500−xD)dx=∫dtwhere C C\,C and D D\,D are positive integers to be found.
(ii) Hence, find t t\,t in terms of xxx.
(iii) Calculate the number of hours it takes for half of the student population to have heard the rumor.