The cables of a suspension bridge are in the shape of a parabola. The
towers supporting the cables are 600 feet apart and 80 feet high. If the
cables touch the road surface midway between the towers, what is the
equation of the cables?
Since the vertex is at (0,0), let the parabola be
y = ax^2
since y(300) = 80, a = 80/300^2
To find the equation of the cables in the shape of a parabola, you can use the vertex form of a parabola equation. The vertex form of a parabola is given by:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
In this case, the towers supporting the cables are 600 feet apart, so the equation for the parabola needs to pass through the points (0, 0) and (600, 0) on the x-axis. The vertex of the parabola will be at the midpoint between these two points.
The x-coordinate of the vertex can be found by taking the average of the x-coordinates of the two given points:
h = (0 + 600) / 2 = 300
Since the cables touch the road surface at the vertex, the y-coordinate of the vertex will be 80 feet.
Therefore, the vertex of the parabola is (300, 80).
We can substitute the vertex into the vertex form equation to find the value of 'a'. Let's use the point (0,0) to get:
0 = a(0 - 300)^2 + 80
0 = a (-300)^2 + 80
0 = 90000a + 80
-90000a = 80
a = -80 / 90000
Simplifying this expression gives: a = -2 / 2250
Now that we have the value of 'a', we can write the equation for the cables in the shape of a parabola:
y = (-2 / 2250)(x - 300)^2 + 80
Therefore, the equation of the cables is y = (-2 / 2250)(x - 300)^2 + 80.