A suspension bridge with weight uniformly distributed along its length has twin towers that extend 75 meters above the road surface and are 400 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 100 meters form the center. (Assume that the road is level)
Can you please explain how you found your answer?
2007-11-26 17:29:02 · 5 answers · asked by BBT 1 in Science & Mathematics ➔ Mathematics
Since the cables are parabolic in shape, for you to find any point along the cable you need to find the equation that defines that parabola. The general equation of a parabola is y=ax^2+bx+c where a,b and c are constants. The vertex of the parabola is (-b/2a;c-b^2/4a). The axis of symmetry of the parabola is the line x= -b/2a. The y-intercept is the point (0,c). The parabola opens upwards if a>0 and dowmwards if a<0. From your info the cables touch the road surface @ centre of bridge i.e. x=200m. Maximum height of cables above road surface y= 75m. You draw your parabola(cable) with the centre of bridge as lowest point (200:0) and twin towers as highest points (0;75)&(400;75). From this -b/2a=200 and c-b^2/4a=0. therefore b^2/4a=c and b=-400a. When x=0, y=75 therefore c=75. Solving for the constants gives a=3/1600; b= -3/4. Hence the parabolic equation is y= 3x^2/1600 - 3x/4 + 75. Solving for x=100m gives y=18.75m. Therefore the height of the cables at a distance 100m from the centre is 18.75m.
2007-11-26 18:47:37 · answer #1 · answered by Trevor P 2 · 0⤊ 0⤋
Let the road surface at the center of the span be (0, 0). Then the towers are at x = -200 and x = +200. At these points, the cables are 75 meters above the road [given], so the equation of the parabola must be y = 75(x/200)^2. Evaluate this at x = +/- 100, and you're done.
2007-11-26 17:36:00 · answer #2 · answered by Anonymous · 0⤊ 0⤋
Using the center of the span as (0,0),
75 = a(200)^2
h = a(100)^2
h/75 = (100/200)^2
h = 75/4 = 18.75 m
2007-11-26 17:50:28 · answer #3 · answered by Helmut 7 · 0⤊ 0⤋
height along y axis
so parabola eqn is x^2=4ay+c
2a=75
x^2=150y+c
now find c by substituting pt(200,0)
now we will get the eqn of parabola
now sub y=100 and find x?
completed.........
2007-11-26 17:37:23 · answer #4 · answered by gnana 2 · 0⤊ 0⤋
uhhh.... all i know is smallest side of triangle =a
2nd smallest=b
largest side =c
and a^2+b^2=c^2
:) im still in algebra!
2007-11-26 17:32:59 · answer #5 · answered by kit-kat 2 · 0⤊ 0⤋