A man wants to create a Japanese garden with a parabolic footbridge to cross the creek in front of his garden. The bridge needs to span 5 meters and - according to the design he has selected - is 2 meters tall at the apex. Find a quadratic model that describes the shape of the bridge.
If we place the vertex at (0,2), then
y = 2-ax^2
Since the span is 5 meters, the roots are at x = ±5/2
So, 2 - 25/4 a = 0
a = 8/25
y = 2 - 0.32x^2
To find a quadratic model that describes the shape of the bridge, we need to consider the general form of a quadratic equation. A quadratic equation is typically expressed as:
f(x) = ax^2 + bx + c
In this case, we want to find a quadratic equation that represents the shape of the bridge. We can assume that the bridge is symmetrical, so we only need to focus on one side of the parabola.
Considering the given information, we can establish the following:
1. The bridge spans 5 meters: This means that the x-axis width of the parabolic curve is 5 meters. However, since the bridge is symmetrical, we can consider the distance from the center of the bridge to either end as half of the width, which is 2.5 meters.
2. The bridge is 2 meters tall at the apex: This indicates that the maximum height of the curve occurs at the center, where x = 0. Therefore, f(0) = 2.
Now, let's determine the values of a, b, and c in the quadratic equation f(x) = ax^2 + bx + c.
1. We know that the vertex of a parabola is given by the equation x = -b/2a. Since our parabola is symmetric around the y-axis, the vertex is at x = 0. Therefore, -b/2a = 0, which implies that b = 0.
2. We can find the value of a by using the information f(0) = 2. Substituting x = 0 into the quadratic equation gives us f(0) = a(0)^2 + 0(0) + c = a(0) + c = c. Therefore, c = 2.
3. To find the value of a, we need another point on the curve. One such point is (2.5, 0), since the end of the bridge (2.5 meters away from the center) is at the same height as the x-axis. Plugging this point into the equation, we get 0 = a(2.5)^2 + 0(2.5) + 2, which simplifies to 6.25a + 2 = 0. Solving for a, we find a = -2/6.25 = -0.32 (approximately).
Therefore, the quadratic equation that describes the shape of the bridge is:
f(x) = -0.32x^2 + 0x + 2
This equation represents the height of the bridge at any given distance from the center.