The general form of a parabola is [tex]y = a(x - h)^2 + k[/tex], where the vertex of the parabola is (h,k).
- The equation of the parabola is [tex]y = 0.0025(x - 90)^2 + 6[/tex].
- The distance between the supports at a vertical height of 26.25 is 180ft
Given that
[tex]y = a(x - h)^2 + k[/tex]
A. The quadratic equation
The lowest point (which stands for the vertex) is given as:
[tex](h,k) = (90,6)[/tex]
So, we have:
[tex]y = a(x - 90)^2 + 6[/tex]
From the question, we understand that:
At a horizontal distance of 30ft, the cable is at 15ft. This means:
[tex](x,y) = (30,15)[/tex]
Substitute these values in: [tex]y = a(x - 90)^2 + 6[/tex]
So, we have:
[tex]15 = a(30 - 90)^2 + 6[/tex]
[tex]15 = a(- 60)^2 + 6[/tex]
[tex]15 = a \times 3600 + 6[/tex]
Collect like terms
[tex]3600a = 15 - 6[/tex]
[tex]3600a = 9[/tex]
Make a, the subject
[tex]a = \frac{9}{3600}[/tex]
[tex]a = 0.0025[/tex]
Substitute [tex]a = 0.0025[/tex] in [tex]y = a(x - 90)^2 + 6[/tex].
[tex]y = 0.0025(x - 90)^2 + 6[/tex]
B. The horizontal distance at a height of 26.25ft.
This means that, y = 26.25
So, we have:
[tex]26.25 = 0.0025(x - 90)^2 + 6[/tex]
Collect like terms
[tex]-6 + 26.25 = 0.0025(x - 90)^2[/tex]
[tex]20.25= 0.0025(x - 90)^2[/tex]
Divide both sides by 0.0025
[tex]8100 = (x - 90)^2[/tex]
Take square roots of both sides
[tex]\±90 = x - 90[/tex]
Solve for x
[tex]x = 90 \± 90[/tex]
To the left of the cable, the horizontal distance is:
[tex]x_1 = 90 - 90[/tex]
[tex]x_1 = 0[/tex]
To the right of the cable, the vertical distance is:
[tex]x_2 = 90 + 90[/tex]
[tex]x_2 = 180[/tex]
So, the coordinates of the left and right support at a height of 26.25 is:
(0,26.25) and (180, 26.25)
The distance (d) between them is calculated using:
[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 -y_1)^2}[/tex]
[tex]d = \sqrt{(180- 0)^2 + (26.25-26.25)^2}[/tex]
[tex]d = \sqrt{180^2 + 0^2}[/tex]
[tex]d = \sqrt{180^2}[/tex]
[tex]d = 180[/tex]
The distance between them is 180ft
Read more about equations of parabola at:
https://brainly.com/question/5428385
