Question

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? ✔ y = 0.0025(x - 90)² + 6 The main cable attaches to the left bridge support at a height of ft.

Answer 1

Answer:

The height of the left bridge is 26.25 feet

Step-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

$y = a(x - h)^2 + k$

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,

$y = a(x - 90)^2 + 6\cdots(i)$

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.

$15 = a(30 - 90)^2 + 6$

$\Rightarrow 15-6=3600a$

$\Rightarrow a= 0.0025.$

Putting the value of constant $a$ in the equation (I) to get the required equation, we have

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

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

$y= 0.0025(0 - 90)^2 + 6$

$y=26.25$

Hence, the height of the left bridge is 26.25 feet.

Answer 2

Answer:

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

26.25ft

180ft

Step-by-step explanation

it is correct.