The main cable of a suspension bridge forms a parabola described by the equation [tex]\( y = a(x - h)^2 + k \)[/tex], where:
- [tex]\( y \)[/tex] is the height in feet of the cable above the roadway,
- [tex]\( x \)[/tex] is the horizontal distance in feet from the left bridge support,
- [tex]\( a \)[/tex] is a constant,
- [tex]\((h, k)\)[/tex] is the vertex of the parabola.

Given:
- At a horizontal distance of [tex]\( 30 \)[/tex] feet, the cable is [tex]\( 15 \)[/tex] feet above the roadway.
- The lowest point of the cable is [tex]\( 6 \)[/tex] feet above the roadway and is at a horizontal distance of [tex]\( 90 \)[/tex] feet from the left bridge support.

Which quadratic equation models the situation correctly?



Answer :

To determine the quadratic equation that models the situation, we start by using the vertex form of a parabolic equation:

[tex]\[ y = a(x - h)^2 + k \][/tex]

Given information:
- The vertex [tex]\((h, k)\)[/tex] of the parabola is [tex]\((90, 6)\)[/tex]. This means the lowest point of the cable is 6 feet above the roadway at 90 feet from the left bridge support.
- At a horizontal distance of 30 feet, the height of the cable, [tex]\(y\)[/tex], is 15 feet.

Substituting the vertex into the standard form gives us:

[tex]\[ y = a(x - 90)^2 + 6 \][/tex]

Next, we need to find the value of [tex]\(a\)[/tex] by using the given point [tex]\((30, 15)\)[/tex]. Substituting [tex]\(x = 30\)[/tex] and [tex]\(y = 15\)[/tex] into the equation, we get:

[tex]\[ 15 = a(30 - 90)^2 + 6 \][/tex]

Simplifying inside the parentheses:

[tex]\[ 15 = a(-60)^2 + 6 \][/tex]

Further simplification of the terms gives:

[tex]\[ 15 = a(3600) + 6 \][/tex]

To isolate [tex]\(a\)[/tex], we subtract 6 from both sides:

[tex]\[ 9 = a(3600) \][/tex]

Now, divide both sides by 3600 to solve for [tex]\(a\)[/tex]:

[tex]\[ a = \frac{9}{3600} = 0.0025 \][/tex]

Thus, the quadratic equation that models this situation is:

[tex]\[ y = 0.0025(x - 90)^2 + 6 \][/tex]

So the correct quadratic equation is:

[tex]\[ y = 0.0025(x - 90)^2 + 6 \][/tex]