Respuesta :
The vertex is (h,k). This is the lowest point of the parabola.
First part.
1) The information that the lowest point of the cable is 6ft above the roadway and at a horizontal distance of 90 ft from the left bridge support, means that those are the coordinates of the vertex:
=> h = 90, k = 6
=> y = a(x - 90)^2 + 6
2) The information that at a horizontal distance of 30 ft, the cable is 15 ft above the roadway =>15 = a (30 - 90)^2 + 6
=> a = [15 - 6] / (-60)^2 = 0.0025
So, the equation of the parabola is y = 0.0025 (x - 90)^2 + 6
Answer: y = 0.0025 (x - 90)^22 + 6
Second part: The main cable attaches to the left bridge support at a height of 26.25 ft.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?
Find x when y = 26.25
26.25 = 0.0025(x - 90)^2
=> (x - 90)^2 = 26.25 / 0.0025 = 10,500
=> x - 90 = +/-√10,500
x - 90 = +/- 102.47
=> x = 90 +/- 102.47
=> x1 = 90 - 102.47 = -12.47
and x2 = 90 + 102.47 = 192.47
Distance = x2 - x1 = 192.47 - (- 12.47) = 204.94
Answer: 204.94
First part.
1) The information that the lowest point of the cable is 6ft above the roadway and at a horizontal distance of 90 ft from the left bridge support, means that those are the coordinates of the vertex:
=> h = 90, k = 6
=> y = a(x - 90)^2 + 6
2) The information that at a horizontal distance of 30 ft, the cable is 15 ft above the roadway =>15 = a (30 - 90)^2 + 6
=> a = [15 - 6] / (-60)^2 = 0.0025
So, the equation of the parabola is y = 0.0025 (x - 90)^2 + 6
Answer: y = 0.0025 (x - 90)^22 + 6
Second part: The main cable attaches to the left bridge support at a height of 26.25 ft.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?
Find x when y = 26.25
26.25 = 0.0025(x - 90)^2
=> (x - 90)^2 = 26.25 / 0.0025 = 10,500
=> x - 90 = +/-√10,500
x - 90 = +/- 102.47
=> x = 90 +/- 102.47
=> x1 = 90 - 102.47 = -12.47
and x2 = 90 + 102.47 = 192.47
Distance = x2 - x1 = 192.47 - (- 12.47) = 204.94
Answer: 204.94
The quadratic equation that models the situation correctly will be [tex]y = 0.0025(x-90)^2+6[/tex] and the distance between the supports will be 180ft and this can be determine by using the arithmetic operations.
Given :
- Parabola - [tex]\rm y = a(x-h)^2+k[/tex]
- 'y' is the height in feet of the cable above the roadway and 'x' is the horizontal distance in feet from the left bridge support.
- 'a' is a constant, and (h, k) is the vertex of the parabola.
- At a horizontal distance of 30 ft, the cable is 15 ft above the roadway.
- The lowest point of the cable is 6ft above the roadway and is a horizontal distance of 90 ft from the left bridge support.
Given that the lowest point of the cable is 6ft above the roadway and is a horizontal distance of 90 ft from the left bridge support means that vertex is (90,6), that is, h = 90 and k = 6.
Now, it is also given that at a horizontal distance of 30 ft, the cable is 15 ft above the roadway, that is, y = 15 and x = 30.
Now, put the values of x, y, h, and k in the equation of parabola.
[tex]15 = a(30-90)^2+6[/tex]
[tex]9=a(-60)^2[/tex]
[tex]a = 2.5\times 10^{-3}[/tex]
Therefore, the quadratic equation that models the situation correctly will be:
[tex]y = 0.0025(x-90)^2+6[/tex]
Given that main cable attaches to the left bridge support at a height of 26.25 ft. The main cable attaches to the right bridge support at the same height as it attaches to the left bridge support then the distance between them will be:
[tex]26.25 = 0.0025(x-90)^2+6[/tex]
[tex]\dfrac{20.25}{0.0025}= (x-90)^2[/tex]
[tex]8100 = (x-90)^2[/tex]
[tex]x^2+8100-180x=8100[/tex]
[tex]x^2-180x=0[/tex]
[tex]x(x-180)= 0[/tex]
x = 0 , 180
Therefore, the distance between the supports will be 180ft.
For more information, refer the link given below:
https://brainly.com/question/5527217
