A landscape architect is designing a decorative concrete partition for a park. The height of the partition, HHH metres, at a horizontal distance ddd metres from a central pillar, is modeled by the equation
H=24d+9,d≥−2.25 H = 2\sqrt{4d + 9}, \quad d \ge -2.25 H=24d+9,d≥−2.25A straight support beam, represented by line lll, is attached to the partition at the point P(4,10)P(4, 10)P(4,10). The beam is perpendicular to the curve of the partition at PPP.
Use calculus to show that an equation for the line lll is
5d+4H−60=0 5d + 4H - 60 = 0 5d+4H−60=0The region RRR is the cross-sectional area of the partition's side, bounded by the curve, the ground (the line H=0H = 0H=0), and the support beam lll.
Use algebraic integration to find the exact area of RRR.
