To determine the distance between the left and right supports of the suspension bridge, we need to analyze the given information and deduce where the cable attaches to the supports.
* Given the quadratic equation of the parabola representing the cable:
[tex]\[
y = 0.0025(x - 90)^2 + 6
\][/tex]
where:
- [tex]\( y \)[/tex] is the height of the cable above the roadway in feet.
- [tex]\( x \)[/tex] is the horizontal distance from the left bridge support in feet.
- The vertex of the parabola [tex]\( (h, k) = (90, 6) \)[/tex] is the lowest point of the cable.
* The cable attaches to the bridge supports at a height of [tex]\( 26.25 \)[/tex] feet.
To find the points where the cable attaches to the supports, we need to determine the values of [tex]\( x \)[/tex] when [tex]\( y = 26.25 \)[/tex].
1. Starting with the equation:
[tex]\[
26.25 = 0.0025(x - 90)^2 + 6
\][/tex]
2. Subtract 6 from both sides to isolate the quadratic term:
[tex]\[
20.25 = 0.0025(x - 90)^2
\][/tex]
3. Divide both sides by 0.0025 to solve for [tex]\((x - 90)^2\)[/tex]:
[tex]\[
8100 = (x - 90)^2
\][/tex]
4. Take the square root of both sides to solve for [tex]\( x - 90 \)[/tex]:
[tex]\[
x - 90 = \pm 90
\][/tex]
5. Solve for [tex]\( x \)[/tex] in both cases to find the points of attachment:
- For [tex]\( x - 90 = 90 \)[/tex]:
[tex]\[
x = 180
\][/tex]
- For [tex]\( x - 90 = -90 \)[/tex]:
[tex]\[
x = 0
\][/tex]
The cable attaches to the left support at [tex]\( x = 0 \)[/tex] and to the right support at [tex]\( x = 180 \)[/tex].
6. Calculate the distance between the supports:
[tex]\[
\text{Distance between supports} = 180 - 0 = 180 \text{ feet}
\][/tex]
Therefore, the distance between the left and right supports is 180 feet.
