A manhole is 12 ft from the centerline of a 30-ft wide street that has a 6-in. parabolic crown. The street center at the station of the manhole is at elevation 612.58 ft. What is the elevation of the manhole cover?
We have a parabola, and the crown is 6 inches over a 30-foot width, i.e. 15' on each side of the crown.
The elevation of the crown is 612.58, and on each side of the road the elevation is 612.58-0.5'=612.08.
Taking x=0 at the crown, the equation of a parabola with the crown (vertex) at (0,612.58) is
y=f(x)=-(0.5/225)x^2+612.58
by fitting a parabola through these three points.
(0,612.58), (-15,612.08), (15,612.08)
So the elevation (of the centre) of the manhole can be evaluated at x=12, or f(12).
Well, if the manhole cover is feeling fancy, it might consider getting raised up on its own pedestal to feel more elevated. But in this case, we'll stick to the facts.
Now, let's tackle this manhole mystery. Since the street has a 6-inch parabolic crown (which I assume is not encrusted with diamonds), we need to find the highest point of the parabola.
Since we already know the width of the street is 30 feet, the distance from the centerline of the street to the side is half of that, which is 15 feet.
So, now we have the height of the parabola, which is 6 inches, or 0.5 feet. We can assume that the highest point of the parabola is at the centerline.
To find the elevation of the manhole cover, we need to determine the vertical distance from the centerline to the manhole. Since the manhole is 12 feet away from the centerline, we can use a little bit of parabolic math to find this distance.
The equation for the parabola is:
y = ax^2
Where 'y' is the height of the parabola and 'x' is the distance from the centerline. We want to find 'a', which is the coefficient that determines the shape of the parabola.
Since the highest point of the parabola is at the centerline, we can use that point to find 'a':
0.5 = a * (0)^2
0.5 = a * 0
a = 0.5
Now that we know 'a', we can find the elevation of the manhole cover. The equation becomes:
y = 0.5x^2
Now, we plug in the distance from the centerline to the manhole, which is 12 feet:
y = 0.5 * (12)^2
y = 0.5 * 144
y = 72 feet
So, the elevation of the manhole cover is 612.58 + 72 = 684.58 feet.
Now just make sure the manhole cover isn't doing high jumps to celebrate its elevated status!
To find the elevation of the manhole cover, we can use the concept of the parabolic crown of the street.
Step 1: Determine the equation of the parabolic crown.
The equation of a parabola with its vertex at the origin can be written as y = ax^2. In this case, the vertex is at the centerline of the street, which means the x-axis is at the centerline.
Since the width of the street is given as 30 ft, the x-coordinate of the right side of the street is x = 15 ft. We also know that at this point, the y-coordinate is 0 (since it is the centerline of the street).
Plugging these values into the equation, we can solve for a:
0 = a(15)^2
0 = 225a
a = 0
So, the equation of the parabolic crown is y = 0.
Step 2: Determine the elevation of the manhole cover.
Since the parabolic crown is a flat line (y = 0), it means that the elevation of the manhole cover will be the same as the elevation of the centerline of the street at the location of the manhole.
Given that the centerline elevation at the manhole station is 612.58 ft, the elevation of the manhole cover is 612.58 ft.
To find the elevation of the manhole cover, we need to consider the parabolic shape of the street's crown and the distance of the manhole from the street center. Here's how you can calculate it step by step:
1. Convert the width of the street into inches:
Since the street width is given as 30 ft, we need to convert it to inches for consistency. Since there are 12 inches in a foot, the width becomes:
Width = 30 ft * 12 in/ft = 360 in.
2. Determine the equation of the parabolic crown:
A parabolic crown is symmetric, so the equation takes the form of y = ax^2, where y represents the elevation, and x represents the distance from the centerline. We know that the maximum elevation is at the centerline, and the elevation gradually decreases as we move away from the centerline. Let's assume the equation is y = ax^2.
3. Find the value of 'a':
We can determine the value of 'a' by using the given information. At the centerline, the elevation is 612.58 ft. Since x = 0 at the centerline, we have:
612.58 ft = a(0)^2
612.58 ft = 0
Therefore, the value of 'a' is irrelevant since any value multiplied by zero is zero.
4. Determine the equation with the value of 'a':
With 'a' being zero, the equation simplifies to:
y = 0x^2
y = 0
5. Calculate the elevation of the manhole cover:
Since the manhole is 12 ft from the centerline, we need to convert it to inches:
Distance from the centerline = 12 ft * 12 in/ft = 144 in
Now, substitute this value into the equation to find the elevation:
y = 0 * (144)^2
y = 0
Therefore, the elevation of the manhole cover is 0 ft or ground level.
In conclusion, the elevation of the manhole cover is 0 ft, which means it is at ground level.