To solve this problem, we follow these steps:
1. Identify Key Information:
- The vertex of the parabola (the lowest point) is given by [tex]$(h, k) = (90, 6)$[/tex]. This means the lowest point is at [tex]$x = 90$[/tex] feet, and its height above the roadway is [tex]$6$[/tex] feet.
- Another point on the parabola is given as [tex]$(x_1, y_1) = (30, 15)$[/tex]. This means when [tex]$x = 30$[/tex] feet, the height of the cable above the roadway is [tex]$15$[/tex] feet.
2. Form of the Parabolic Equation:
- The general quadratic equation for the parabola is [tex]\( y = a(x - h)^2 + k \)[/tex].
- Substituting the vertex [tex]$(h, k)$[/tex] into the equation, we get [tex]\( y = a(x - 90)^2 + 6 \)[/tex].
3. Determine the Constant [tex]\( a \)[/tex]:
- We know that when [tex]$x = 30$[/tex], [tex]$y = 15$[/tex], so we substitute these values into the equation to find [tex]\( a \)[/tex]:
[tex]\[
15 = a(30 - 90)^2 + 6
\][/tex]
- Simplify the equation to find [tex]\( a \)[/tex]:
[tex]\[
15 = a \cdot 3600 + 6
\][/tex]
[tex]\[
15 - 6 = 3600a
\][/tex]
[tex]\[
9 = 3600a
\][/tex]
[tex]\[
a = \frac{9}{3600} = 0.0025
\][/tex]
4. Parabolic Equation:
- With [tex]\( a = 0.0025 \)[/tex], the equation of the parabola becomes:
[tex]\[
y = 0.0025(x - 90)^2 + 6
\][/tex]
5. Height at the Left Bridge Support:
- To find the height at the left bridge support, we substitute [tex]$x = 0$[/tex] into the equation:
[tex]\[
y = 0.0025(0 - 90)^2 + 6
\][/tex]
- Simplify the expression to find the height:
[tex]\[
y = 0.0025 \cdot 8100 + 6
\][/tex]
[tex]\[
y = 20.25 + 6
\][/tex]
[tex]\[
y = 26.25
\][/tex]
Therefore, the height of the main cable where it attaches to the left bridge support is [tex]\( 26.25 \)[/tex] feet.
