28 8. A community is concerned about the rising level of pollutant in its local pond and applies a chemical treatment to stop the increase of pollutant. The concentration, x parts per million (ppm), of the pollutant in the pond water t days after the chemical treatment was applied, is modelled by the differential equation d d x t t x t x t = + − 3 3 1 3 2 cosh cosh tanh ( )I When the chemical treatment was applied the concentration of pollutant was 3 ppm. (a) Use the iteration formula d d y x y y h n n n     ≈ − + ( ) 1 once to estimate the concentration of the pollutant in the pond water 6 hours after the chemical treatment was applied. (4) (b) Show that the transformation u = x 3 transforms the differential equation (I) into the differential equation d d u t u t t + = + tanh cosh ( ) 1 3 II (3) (c) Determine the general solution of equation (II) (4) (d) Hence find an equation for the concentration of pollutant in the pond water t days after the chemical treatment was applied. (3) (e) Find the percentage error of the estimate found in part (a) compared to the value predicted by the model, stating if it is an overestimate or an underestimate. (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________