a parabolic shaped bridge is 30m wide. the height of the bridge is 5m 8m from the edge of the bridge.
a) what is the equation that models the bridge?
b)what is the maximum height of the bridge?
a) To model the parabolic shaped bridge, we can use the equation of a parabola in vertex form: y = a(x-h)^2 + k, where (h,k) represents the vertex of the parabola.
Since the bridge is symmetric, the vertex will lie at the midpoint of the bridge's width, which is at x = 0.5 * 30 = 15 meters. The height of the bridge is given as 5 meters at x = 8 meters from the edge, which means the vertex has a height of 5 meters.
So, plugging in the values of h = 15 and k = 5 into the vertex form equation, we have:
y = a(x-15)^2 + 5
b) To find the maximum height of the bridge, we need to find the value of a. We can use the additional information given, which is that the height of the bridge is 5 meters at x = 8 meters from the edge.
Plugging these values into the equation, we have:
5 = a(8-15)^2 + 5
0 = a(-7)^2
0 = 49a
Since 49a = 0, we know that a = 0.
Therefore, the equation that models the bridge is y = 5, which means the bridge is a flat line at a height of 5 meters. Thus, the maximum height of the bridge is 5 meters.