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.
:
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 = {{{200/320000}}}
a = .000625
:
Find be substitute .000625 for a in the 1st equation:
160000(.000625) + 400b = -100
100 + 400b = -100
400b = -100 -100
b = {{{(-200)/400}}}
b =-.5
:
The equation: y = .000625x^2 - .5x + 100 
Plot this graph, looks like what we would expect.
{{{ graph( 300, 200, -200, 900, -20, 120, .000625x^2-.5x+100) }}}
:
 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