The shape of the cable between the towers of a suspension bridge is a parabola. We are given that the towers are 800ft apart and rise 160ft above the road. The cable touches the sides of the road midway between the towers. We need to find the height of the cable 100ft from the towers.
Since the cable touches the sides of the road midway between the towers, we can assume that the lowest point of the cable is at the midpoint between the towers. This means that the vertex of the parabola is at the midpoint between the towers, and the axis of symmetry is a vertical line passing through this point.
To find the height of the cable 100ft from the towers, we need to find the value of y when x = 100.
Let's consider the coordinates of the vertex of the parabola. The x-coordinate of the vertex is the average of the x-coordinates of the towers, which is (0 + 800) / 2 = 400. The y-coordinate of the vertex is the height of the towers above the road, which is 160ft.
Now let's find the equation of the parabola in vertex form. The equation of a parabola in vertex form is y = a(x - h)^2 + k, where (h, k) are the coordinates of the vertex. In this case, h = 400 and k = 160.
Substituting these values into the equation, we get y = a(x - 400)^2 + 160.
To find the value of a, we can use the fact that the cable touches the sides of the road at x = 0 and x = 800. Substituting these values into the equation, we get y = a(0 - 400)^2 + 160 and y = a(800 - 400)^2 + 160.
Simplifying these equations, we get 160 = a(400)^2 + 160 and 160 = a(400)^2 + 160.
Solving these equations, we find that a = -0.0001.
Now we can find the height of the cable 100ft from the towers. Substituting x = 100 into the equation y = -0.0001(x - 400)^2 + 160, we get y = -0.0001(100 - 400)^2 + 160.
Simplifying this equation, we get y = -0.0001(300)^2 + 160.
Calculating this expression, we find that y = -0.0001(90000) + 160 = -9 + 160 = 151.
Therefore, the height of the cable 100ft from the towers is 151ft.