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) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________
Get full Socratic AI guidance on this question — free in the Applaa desktop app
Appy Buddy guides you step-by-step toward the answer without giving it away. Type your attempt and get instant, mark-scheme-aware clues that teach you to think like an examiner.
Join Applaa Community
Create your own games, learn AI concepts, program interactive apps, and share with a kid-safe community approved by parents. Free forever on Windows and Mac.
Available for Windows and macOS · COPPA Compliant
Exam Specification Info
This question is part of the UK A-Level Mathematics syllabus. In the actual exam, structured questions typically require linking specific keywords to gain full marks. Applaa helps you drill these topics.
While Other Kids Get an AI Head Start, Is Yours Falling Behind?
Give your child the same AI app-building, exam prep and tutoring edge — free for the first month, no credit card needed.
Join 10,000+ students building their first AI-powered apps with Applaa