the main cable of a suspension bridge forms a parabola, described by the equation $y = a(x - h)^2 + k$…

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.\n\nat 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.\n\nwhich quadratic equation models the situation correctly?\n$y = -0.0025(x - 90)^2 + 6$\n$y = -0.0025(x - 30)^2 + 15$\n$y = 0.0025(x - 90)^2 + 6$\n$y = 0.0025(x - 30)^2 + 15$

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.\n\nat 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.\n\nwhich quadratic equation models the situation correctly?\n$y = -0.0025(x - 90)^2 + 6$\n$y = -0.0025(x - 30)^2 + 15$\n$y = 0.0025(x - 90)^2 + 6$\n$y = 0.0025(x - 30)^2 + 15$

Answer

Explanation:

Step1: Identify vertex (h,k)

The lowest point (vertex) is at $x=90$, $y=6$, so $(h,k)=(90,6)$. Substitute into $y=a(x-h)^2+k$: $y=a(x-90)^2+6$

Step2: Solve for constant a

Use point $(x,y)=(30,15)$: $15=a(30-90)^2+6$ $15=a(-60)^2+6$ $15=3600a+6$ $3600a=15-6=9$ $a=\frac{9}{3600}=0.0025$

Step3: Write final equation

Substitute $a=0.0025$ into the vertex form: $y=0.0025(x-90)^2+6$

Answer:

$y = 0.0025(x - 90)^2 + 6$