SOLUTION: The cable for a suspension bridge is shaped like a parabola. The towers are 100 ft tall and 800 ft apart. It touches the roadway at the center. How long are the support cables at 5

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: The cable for a suspension bridge is shaped like a parabola. The towers are 100 ft tall and 800 ft apart. It touches the roadway at the center. How long are the support cables at 5      Log On


   

Question 189085: The cable for a suspension bridge is shaped like a parabola. The towers are 100 ft tall and 800 ft apart. It touches the roadway at the center. How long are the support cables at 50 ft, 200 ft, and 350 ft from the center?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
The cable for a suspension bridge is shaped like a parabola.
The towers are 100 ft tall and 800 ft apart.
It touches the roadway at the center.
:
determine the quadratic equation: y = ax^2 + bx + c
:
From the information given we have 3 coordinates:
x=400, y=0; where it touches the roadway in the center
x=0, y=100; left tower; this also the y intercept, therefore c = 100
x=800, y=100; right tower
:
Solve for a & b
x=400, y=0
400^2a + 400b + 100 = 0
160000a + 400b = -100
:
x=800, y=100
800^2a + 800b + 100 = 100
640000a + 800b = 100 - 100
640000a + 800b = 0
:
Multiply the 1st equation by 2 and subtract from the above equation
640000a + 800b = 0
320000a + 800b = -200
-----------------------subtraction eliminates b
320000a = +200
a = 200%2F320000
a = .000625
:
Find be substitute .000625 for a in the 1st equation:
160000(.000625) + 400b = -100
100 + 400b = -100
400b = -100 -100
b = %28-200%29%2F400
b =-.5
:
The equation: y = .000625x^2 - .5x + 100
Plot this graph, looks like what we would expect.
+graph%28+300%2C+200%2C+-200%2C+900%2C+-20%2C+120%2C+.000625x%5E2-.5x%2B100%29+
:
How long are the support cables at 50 ft from the center
that means: 400 - 50 = 350; x=350. find y:
y = .000625(350^2) - .5(350) + 100
y = .000625(122500) - 175 + 100
y = 1.5625 ft
:
How long are the support cables at 200 ft from the center
that means: 400 - 200 = 200; x=200. find y:
y = .000625(200^2) - .5(200) + 100
y = .000625(40000) - 100 + 100
y = 25 ft
:
How long are the support cables at 350 ft from the center
that means: 400 - 350 = 50; x=50. find y:
y = .000625(50^2) - .5(50) + 100
y = .000625(2500) - 25 + 100
y = 76.5625 ft