28 8. Two different colours of paint are being mixed together in a container. The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container. Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint. The colour of the paint mixture is now altered by • adding red paint to the container at a rate of 2 litres per second • adding blue paint to the container at a rate of 1 litre per second • pumping fully mixed paint from the container at a rate of 3 litres per second. Let r litres be the amount of red paint in the container at time t seconds after the colour of the paint mixture starts to be altered. (a) Show that the amount of red paint in the container can be modelled by the differential equation d 2 d r r t α = − where α is a positive constant to be determined. (2) (b) By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second. (6) It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint. (c) Use this information to evaluate the model, giving a reason for your answer. (1) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________