The cross-section of a parabolic archway is modeled by the curve C C\,C with equation y=6x−x2y = 6x - x^2y=6x−x2, where y y\,y is the vertical height in metres and x x\,x is the horizontal distance from the left-hand base O O\,O at the origin. A straight support strut L L\,L is to be installed from the origin to a point Q Q\,Q on the archway. The strut follows the equation y=kxy = kxy=kx, where k k\,k is a constant and 0<k<60 < k < 60<k<6.
Find, in terms of kkk, the coordinates of QQQ.
The region R1 R_1\,R1 is the area of the archway cross-section that lies above the support strut and below the curve CCC.
Show that the area of R1 R_1\,R1 is
(6−k)36 \frac{(6 - k)^3}{6} 6(6−k)3The region R2 R_2\,R2 is the area of the archway cross-section that lies below the support strut, above the horizontal ground (xxx-axis), and is bounded to the right by the archway.
Given that the area of R1 R_1\,R1 is equal to the area of R2R_2R2, find 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.
