The rate of a chemical reaction, rrr grams per minute, is modelled by the function
r=9t−t1.5,t≥0 r = 9t - t^{1.5}, \quad t \ge 0 r=9t−t1.5,t≥0where ttt is the time in minutes since the reaction began. A plot of this rate shows a curve starting at the origin, reaching a peak production rate at the point MMM, and then falling back to zero at the point PPP.
Using calculus, find the coordinates of the maximum point MMM.
The reaction concludes when the rate returns to zero at point PPP. Use algebra to find the ttt-coordinate of PPP.
The finite region RRR is bounded by the curve, the horizontal line passing through MMM, and the vertical line passing through PPP.
Use algebraic integration to find the area of RRR, giving your answer to two decimal places.
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.