An artist is designing a stained-glass panel. The boundary of the panel is modeled by the curve CCC with equation y=6x−x2y = 6x - x^2y=6x−x2, for x≥0x \ge 0x≥0. A decorative lead strip is placed along the line lll with equation y=kxy = kxy=kx, where kkk is a constant and 0<k<60 < k < 60<k<6.
The line lll and the curve CCC intersect at the origin OOO and at the point PPP.
Find, in terms of kkk, the coordinates of PPP.
The region R1R_1R1 is bounded by the curve CCC and the line lll. Show that the area of R1R_1R1 is
(6−k)36 \frac{(6 - k)^3}{6} 6(6−k)3The region R2R_2R2 is bounded by the curve CCC, the xxx-axis, and the line lll, such that R1R_1R1 and R2R_2R2 together comprise the total area under the curve for 0≤x≤60 \le x \le 60≤x≤6. Given that the area of R1R_1R1 is equal to the area of R2R_2R2, find the exact value of kkk.