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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, and [tex]\( (h, k) \)[/tex] is the vertex of the parabola.

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

1. Which quadratic equation models the situation correctly?

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

2. The main cable attaches to the left bridge support at a height of [tex]\( 26.25 \)[/tex] ft.

3. The main cable attaches to the right bridge support at the same height as it attaches to the left bridge support. What is the distance between the supports?



Answer :

GinnyAnswer
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.

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