How Do Parabolic Roads Drain Water?

[darshanpatel]

Homework Statement

Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road is that is 32 feet wide is .4 feet higher in in the center then on the sides.

a) Find an equation if the parabola that models the road surface. (Assume that the orgin is at the center of the road.)

b) How far from the center of the road is the road surface .1 feet lower then in the middle?

Homework Equations

-None-

The Attempt at a Solution

x^2=4py
x^2= 4(.4)y
x^2=1.6y

^part (a) Correct?

x^2=1.6(.3)
x^2=.48
x = .693 feet away from the center

^Part (b) -Correct?

I kind of have a basis but I am a little wary about the answers.

 

[Dick]

The form x^2=4py is fine. If the origin is the center of the road then a point at the center of the road is x=0, y=0 and x is the distance from the center of the road and y is the elevation of the road. What should y be if x is 16?

 

[darshanpatel]

y=.4

but what next?

 

[Dick]

y=.4

but what next?

Use that to solve for p. p isn't equal to 0.4.

 

[darshanpatel]

so like x^2=4py

x^2=4p(.4)
(1/.4)x^2=4p

4p=1/.4
p=1/1.6

x^2=4(1/1.6)y
x^2=6.4y

is that right?
becuase the first time i thought that (0, .4) was the focus, and p=.4 from the focus...

 

[Dick]

so like x^2=4py

x^2=4p(.4)
(1/.4)x^2=4p

4p=1/.4
p=1/1.6

x^2=4(1/1.6)y
x^2=6.4y

is that right?
becuase the first time i thought that (0, .4) was the focus, and p=.4 from the focus...

(1/.4)x^2=4p. x=16. What's p? It's not 1.6 either.

 

[darshanpatel]

p=160

so the equation for the graph would be x^2=640y?

and plugging in .3 for y
would get me answer to part (b)
13.856 feet from center?

 

[Dick]

p=160

so the equation for the graph would be x^2=640y?

and plugging in .3 for y
would get me answer to part (b)
13.856 feet from center?

Yes for the first part. .4 is how far the road is below the center at x=16. The problem is asking "How far from the center of the road is the road surface .1 feet lower then in the middle?". I wouldn't plug in .3 for y.

 

[darshanpatel]

oh, i was thinking of like where is the road only .3 feet or whatever,

do i plug .1 in?

Solved it for .1 and got 8

 

[Dick]

oh, i was thinking of like where is the road only .3 feet or whatever,

do i plug .1 in?

Solved it for .1 and got 8

I really don't like responses like "do i plug .1 in?". Do you or don't you? Sketch a picture of the road surface and tell me.

 

[darshanpatel]

What do you mean you don't like them? I wasn't trying to be rude or anything.
You do plug in .1

 

[Dick]

What do you mean you don't like them? I wasn't trying to be rude or anything.
You do plug in .1

I didn't think you were trying to be rude or anything. I just like hearing "You do plug in .1" better than the "?" thing. Yes, you are right. And for the equation of the road surface they might like x^2=(-640)y better than x^2=640y. x^2=640y is sort of 'upside down'.

 

[darshanpatel]

x^2=640y is upside down?

I don't understand.

 

[Mark44]

x^2=640y is upside down?

I don't understand.

It's upside down relative to the road surface. The parabola above opens up - you want a parabola that opens down, so that water will drain off the road. That's what Dick meant by "upside down."

 

[darshanpatel]

ohhhhhhhhhhhhhhhhhhhhh, thank you, that makes a lot more sense, how would you show the work so that the number comes out to x^2=-640y?

Would you use -.4 when making the equation becuase it is 'upside down' now?

But would it matter because the answer comes out to the same thing no matter what?

 

[Mark44]

ohhhhhhhhhhhhhhhhhhhhh, thank you, that makes a lot more sense, how would you show the work so that the number comes out to x^2=-640y?

Would you use -.4 when making the equation becuase it is 'upside down' now?

Start the work you did earlier with x² = -4py (with p > 0) to get a parabola that opens down.

But would it matter because the answer comes out to the same thing no matter what?

 

[darshanpatel]

Can you help me with how the graph would look?

It's not for my homework, its just so I can understand it.

How would the parabola open and where?

Also to get part (b)

would you plug in -.1 for y and solve?

Becuase when I just plugged in .1 it would give me a negative answer

 

[Mark44]

Can you help me with how the graph would look?

How do you think the graph should look? You want the water to drain off the road, so it should have the high point at the center of the road's cross section.

This means you want a parabola that opens down. If the cross section of the road looked like a parabola that opened up, water would collect in the middle of the road, which isn't good.

Do as Dick suggested many posts ago by putting the vertex of the parabola at the origin. The left and right edges of the road will be at (-16, -.4) and (16, -.4).

If you substitute the x and y values of either point into x² = -4py, you should be able to solve for p, which will be a positive number.

It's not for my homework, its just so I can understand it.

How would the parabola open and where?

Also to get part (b)

would you plug in -.1 for y and solve?

Becuase when I just plugged in .1 it would give me a negative answer

If the high point of the road is at (0, 0), every other point will have y values that are negative. You should get two x-values when y = -.1.