Question 726659: the sydney harbour bridge has its main steel arch in the shape of a parabola. The longest vertical support, situated 135m from its origin is 182.25m high. The road is situated 50m above the base of the pylon. There are 19 vertical steel cables equally spaced between the road and the arch. Fins the equation of the parabolic arch in the form y=ax2 + bx
Answer by DrBeeee(684)   (Show Source):
You can put this solution on YOUR website! Let's place your (x,y) coordinate axis at the apex of the arch. Then we have, to start with
(1) y = ax^2
To find a we substitute the point (135,-182.25) into (1) to get
(2) -182.82 = a*(135)^2 or
(3) a = -182.25/((135)^@) or
(4) a = -0.01
Then we have
(5) y = -0.01x^2
This gives the equation of the parabola (arch) as if it started at the apex or maximum value of the parabola. However the description of the arch has x starting at the base which is 135m to left of (5). Therefore we must shift (5) to the right by 135 and get
(6) y = -0.01(x-135)^2
It also states that y (the height of the arch) is 182.25m above the starting pylon. Therefore we must also shift (5) up by 182.25m, giving us
(7) y = -0.01(x-135)^2 + 182.25
Simplifying (7) we get
(8) y = -0.01(x^2 - 2*135x + 135^2) + 182.25 or
(9) y = -0.01*x^2 + 2*1.35x - 0.01*135^2 + 182.25 or
(10) y = -0.01*x^2 + 2.70x - 0.01*135^2 + 182.25 or
(11) y = -0.01*x^2 + 2.70x
Comparing the coefficient of (11) to
(12) y = ax^2 + bx gives
(13) a = -0.01 and b = 2.70
Let's check the equation of (11).
Is ( 0 = -0.01*0^2 + 0)?
Is (0 = 0)? Yes
Is (182.25 = -0.01*135^2 + 2.70*135)?
Is (182.25 = -182.25 + 364.5)?
Is (182.25 = 182.25)? Yes
Is (0 = -0.01*270^2 + 2.70*270)?
Is (0 = -729 + 729)?
Is (0 = 0)? Yes
Answer: The equation of the parabolic arch is y = -0.01*x^2 +2.70x.
PS The 50m and 19 steel cables do not enter into the problem.
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