The main cable of a suspension bridge forms a parabola, described by the equation y = a(x - h)2 + k, where y is the height in feet of the cable above the roadway, x is the horizontal distance in feet from the left bridge support, a is a constant, and (h, k) is the vertex of the parabola. At a horizontal distance of 30 ft, the cable is 15 ft above the roadway. The lowest point of the cable is 6ft above the roadway and is a horizontal distance of 90 ft from the left bridge support.Which quadratic equation models the situation correctly? The main cable attaches to the left bridge support at a height of ft.The main cable attaches to the right bridge support at the same height as it attaches to the left bridge support. What is the distance between the supports?

Answer:The height of the left bridge is 26.25 feetStep-by-step explanation:Let the left base of the bridge is at the origin and the x-axis represents the roadways as shown in the figure.The given relationship between the variables x and y is[tex]y = a(x - h)^2 + k[/tex]where x is the horizontal distance from the left bridge support and y is the height of the cable above the roadway, (h,k) is the vertex of the parabola, and a is constant.The vertex (h,k) of the parabola is the lowest point of the cable bridge.As the lowest point of the cable is 6ft above the roadway and is a horizontal distance of 90 ft from the left bridge support, so, h=90 and k=6.The given equation become,[tex]y = a(x - 90)^2 + 6\cdots(i)[/tex]At a horizontal distance of 30 ft, the cable is 15 ft above the roadway, so put x=30 and y=15 in the equation (i) to the value of  constant a.[tex]15 = a(30 - 90)^2 + 6[/tex][tex]\Rightarrow 15-6=3600a[/tex][tex]\Rightarrow a= 0.0025.[/tex]Putting the value of constant [tex]a[/tex] in the equation (I) to get the required equation, we have[tex]y = 0.0025(x - 90)^2 + 6[/tex]As the left bridge is at the origin, so the the height of the left bridge is the value of y at origin.Hence, put x=0 in the obtained equation to get the height of the bridge at the left side, we have[tex]y= 0.0025(0 - 90)^2 + 6[/tex][tex]y=26.25[/tex]Hence, the height of the left bridge is 26.25 feet.