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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.