A parabolic arch bridge has a 60ft base and a height of 24ft. Find the height of the arch at a distance of 5, 10, and 20 ft from the center
let the vertex of the parabola be (0,60)
its equation would be
y = ax^2 + 60 , but (30,0) and (-30,0) would also lie on it
0 = a(900) + 60
a = -60/900 = -1/15
equation of bridge is
y = (-1/15)x^2 + 60
so when x = 5
y = (-1/15)(25) + 60 = 175/3 ft
or 58 1/3 ft
or appr 58.33 ft
repeat for x = 10 and x = 20
To find the height of the arch at a distance of 5, 10, and 20 ft from the center, we will use the equation of a parabola in vertex form:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola. In this case, the vertex represents the highest point of the arch, so the vertex is at the center of the bridge.
Given:
Base (distance between the endpoints of the arch) = 60 ft
Height at the vertex = 24 ft
Step 1: Find the vertex
Since the vertex is at the center of the bridge, it is halfway between the endpoints, which gives a distance of 60 ft / 2 = 30 ft from either endpoint.
Therefore, the vertex is at (30, 24).
Step 2: Determine the value of 'a'
Using the vertex coordinates (30, 24) and the point on the arch (0, 0), we can find the value of 'a'.
Plugging in the values into the equation, we get:
0 = a(0 - 30)^2 + 24
0 = 900a + 24
900a = -24
a = -24/900
a = -1/37.5
Step 3: Find the height at a distance of 5 ft from the center
Using the equation of a parabola, we plug in x = 5:
y = (-1/37.5)(5 - 30)^2 + 24
y = (-1/37.5)(-25)^2 + 24
y = (-1/37.5)(625) + 24
y ≈ -16.67 + 24
y ≈ 7.33 ft
Therefore, the height of the arch at a distance of 5 ft from the center is approximately 7.33 ft.
Step 4: Find the height at a distance of 10 ft from the center
Using the equation of a parabola, we plug in x = 10:
y = (-1/37.5)(10 - 30)^2 + 24
y = (-1/37.5)(-20)^2 + 24
y = (-1/37.5)(400) + 24
y ≈ -10.67 + 24
y ≈ 13.33 ft
Therefore, the height of the arch at a distance of 10 ft from the center is approximately 13.33 ft.
Step 5: Find the height at a distance of 20 ft from the center
Using the equation of a parabola, we plug in x = 20:
y = (-1/37.5)(20 - 30)^2 + 24
y = (-1/37.5)(-10)^2 + 24
y = (-1/37.5)(100) + 24
y ≈ -2.67 + 24
y ≈ 21.33 ft
Therefore, the height of the arch at a distance of 20 ft from the center is approximately 21.33 ft.
To find the height of the arch at a specific distance from the center, we can use the equation of a parabola.
The equation of a parabola can be written in the form y = ax^2 + bx + c, where x is the distance from the center of the parabolic arch, and y is the height of the arch at that distance.
We are given that the base of the arch is 60ft, which means the maximum height occurs at the center of the arch.
To find the equation of the parabola, we need to find the values of a, b, and c.
Step 1: Determine the vertex of the parabolic arch
The vertex of the parabolic arch is the point where the maximum height occurs. Since the base of the arch is 60ft, the x-coordinate of the vertex is halfway between 0 and 60, which is 30.
To find the y-coordinate of the vertex, we can substitute the x-coordinate into the equation of the parabola. Since the maximum height is 24ft, we have:
24 = a(30)^2 + b(30) + c
Step 2: Find two additional points on the parabolic arch
To find the height of the arch at a distance of 5ft from the center, we can substitute x = 5 into the equation of the parabola.
Similarly, to find the height of the arch at a distance of 10ft and 20ft from the center, we can substitute x = 10 and x = 20 respectively.
Step 3: Solve the system of equations
The three points we have are (5, y1), (10, y2), and (20, y3), where y1, y2, and y3 are the heights of the arch at those distances from the center.
Substituting these points into the equation of the parabola, we get a system of three equations with three unknowns (a, b, c). We can solve this system of equations to find the values of a, b, and c.
Step 4: Substitute the values of a, b, and c into the equation for a given distance
Once we have found the values of a, b, and c, we can substitute them back into the equation of the parabola along with the desired distance from the center to find the height of the arch at that distance.
By following these steps, we can find the height of the arch at a distance of 5, 10, and 20ft from the center.