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.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.