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]
