The entrance way of the city of North Bay used to be an arch that can be modelled by the equation of parabola. If the edge of the arch is the origin, and the arch is 30 m wide, what is the equation of the parabola if the height of the arch 4 m from the edge of the base is 9 m? Include a drawing in your answer.
To determine the equation of the parabola, we need to find the vertex of the parabola and the equation in the general form (y = ax^2 + bx + c).
Given that the origin (0, 0) is at the edge of the arch and the arch is 30 m wide, we know that the arch's width is from x = -15 to x = 15.
Since the vertex of the parabola is the highest point on the arch, we also know that the vertex lies on the axis of symmetry, which is the line x = 0.
From the given information, when x = 4 (4 m from the edge of the base), the height of the arch is y = 9.
So, the vertex of the parabola is at (0, 9).
Let's use the vertex form of the parabola to find the equation: y = a(x - h)^2 + k, where (h, k) is the vertex.
Plugging in the values of the vertex, we have:
y = a(x - 0)^2 + 9
To find the value of "a", we can use one of the given points on the arch. We'll use the point (4, 9).
9 = a(4 - 0)^2 + 9
0 = 16a
a = 0
Since a = 0, the equation simplifies to:
y = 9
Therefore, the equation of the parabola modeling the entrance way arch is y = 9.
Below is a simple drawing depicting the arch:
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