A landscape architect is designing a feature for a park. The cross-sectional elevation of the terrain is modelled by a curve CCC with equation
y=x12(15−x),x≥0 y = x^{\frac{1}{2}}(15 - x), \quad x \ge 0 y=x21(15−x),x≥0where yyy is the height in decametres and xxx is the horizontal distance in decametres. The curve CCC has a stationary point at PPP.
Using calculus, determine the xxx-coordinate of PPP.
The region R1R_1R1, representing a hill section, is bounded by the curve CCC and the xxx-axis.
The region R2R_2R2, representing a valley section, is bounded by the curve CCC, the xxx-axis, and the vertical line x=kx = kx=k, where k>15k > 15k>15 is a constant.
Given that the area of the hill R1R_1R1 is equal to the area of the valley R2R_2R2, find, using calculus, the exact value of kkk.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions 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, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
