The profile of a decorative architectural arch, as shown in the cross-section of a building design, is modeled using a coordinate system where the vertical height HHH (in decametres) is given by the equation H=x2−9x+22H = x^2 - 9x + 22H=x2−9x+22, where x x\,x is the horizontal distance from a vertical support wall.
The arch is supported by a horizontal beam at a constant height of H=4H = 4H=4 decametres, which we will call line LLL.
The arch intersects the support wall at point DDD.
Write down the coordinates of point DDD.
The arch intersects the horizontal beam L L\,L at the points E E\,E and FFF, as shown.
Find the xxx-coordinate of E E\,E and the xxx-coordinate of FFF.
Two specific design regions are identified:
Given that Area of R1Area of R2=k\displaystyle \frac{\text{Area of } R_1}{\text{Area of } R_2} = kArea of R2Area of R1=k,
use algebraic integration to find the exact value of kkk, giving your answer as a simplified fraction.