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.
