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.
864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.