The underside of a concrete bridge forms a parabolic arch that is 32 meters wide at the water level and twelve meters high in the center. The road (top of the bridge) is forty eight meters wide, and the minimum thickness of the concrete is two meters.
a road that is 48 meters wide? Man, that's some highway!
And, after all the data, just what is the question? The volume of concrete needed?
To find the equation of the parabolic arch formed by the underside of the concrete bridge, we can use the vertex form of a parabola equation:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Given:
Width of the arch at the water level = 32 meters
Height of the arch at the center = 12 meters
Width of the road (top of the bridge) = 48 meters
Minimum thickness of the concrete = 2 meters
1. Finding the vertex:
Since the center of the parabolic arch is at the highest point, its vertex is at (0, 12).
2. Determining the value of "a":
To find the value of "a," we need another point on the curve. At the water level, the width of the arch is 32 meters. Since the center is 0 meters wide, the distance on either side is half of the width, which is 16 meters.
Using the vertex form of the parabola equation, let's substitute the known values:
12 = a(16 - 0)^2 + 12
Subtracting 12 from both sides:
0 = a(16)^2
0 = 256a
Since a cannot be zero, this implies that 0 = 0, and any value of a would work.
3. Writing the equation of the parabolic arch:
Using the vertex form equation and the value of a, we have:
y = a(x - h)^2 + k
y = ax^2 + 12
Thus, the equation of the parabolic arch formed by the underside of the concrete bridge is y = ax^2 + 12, where a can be any non-zero real number.
To find the shape of the parabolic arch formed by the underside of the concrete 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,
- a is a constant that affects the steepness and direction of the parabola, and
- x and y are the coordinates on the graph.
Given that the parabolic arch is 32 meters wide at the water level (bottom of the arch) and twelve meters high in the center, we can determine the vertex of the parabola. The vertex can be found by using the midpoint of the water level and the center height, which are both given:
Vertex: (h, k) = (0, 12)
Next, we need to find the value of a. To do this, we use another point on the parabola. The road width of the bridge is 48 meters, so let's use the midpoint of the road width (24 meters) as another point on the parabola. Using this point, we can solve for a:
x = 24, y = 0 (since the parabola intersects the x-axis at the road level)
0 = a(24 - 0)^2 + 12
0 = 576a + 12
576a = -12
a = -12/576
a ≈ -0.0208
Now we have all the information to write the equation of the parabolic arch:
y = -0.0208(x - 0)^2 + 12
Simplifying further, we get:
y = -0.0208x^2 + 12
So, the equation of the parabolic arch formed by the underside of the concrete bridge is y = -0.0208x^2 + 12.