Exciting riverbed problem

  • Thread starter jrand26
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In summary: A) The area of the rectangle is\int_{-3/2}^{3/2}2- ax^2 dx= (1/2)\int_{-3/2}^{3/2}h- ax^2 dx= 225cm^2. B) The area of the parabola isy=A(x-h)y=-A(x-h)y=-A(x-h)h=-A(x-h)h=-A(x-h)This gives you the area of the parabola, which is 125cm^2.
  • #1
jrand26
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Homework Statement


The cross-section of a channel is parabolic. It is 3 metres wide at the top and 2 metres deep. Find the depth of water, to the nearest cm, when the channel is half full.


Homework Equations


Nil.

The Attempt at a Solution


I found the function for the graph to be [tex]y=x^2-3x[/tex]. Then I figured that I could just plug (0,0) and (3,0) into the integrand, [tex]\frac{\1}{3}x^3-1.5x^2[/tex], and then divide by two to get the answer. I'm pretty sure that the problem lies with the method, not my math. I have a feeling that's not the above method does not solve the problem, but I'm not sure why that is so.

For reference, my method gets 225cm while the book has 126cm.
 
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  • #2
Well, I would have personally interpreted the graph to be such that the parabola passed through the points (-1.5, 2) (0,0) and (1.5, 2). With those points we can find the equation of the parabola, sub in y=1 and solve for x.

Your equations has a width of 3, but its depth is not 2 as the description states.
 
  • #3
Ok thanks, now I'm getting 137cm, and its probably because my function is a bit out of the points. Also, Is my latex correct, I'm supposed to see the [tex] stuff right?
 
  • #4
To start tex code, use [ tex ] but to end it its [ /tex ], without the spaces of course.
 
  • #5
jrand26 said:

Homework Statement


The cross-section of a channel is parabolic. It is 3 metres wide at the top and 2 metres deep. Find the depth of water, to the nearest cm, when the channel is half full.


Homework Equations


Nil.

The Attempt at a Solution


I found the function for the graph to be [tex]y=x^2-3x[/tex]. Then I figured that I could just plug (0,0) and (3,0) into the integrand, [tex]\frac{\1}{3}x^3-1.5x^2[/tex], and then divide by two to get the answer. I'm pretty sure that the problem lies with the method, not my math. I have a feeling that's not the above method does not solve the problem, but I'm not sure why that is so.

For reference, my method gets 225cm while the book has 126cm.
I have corrected the LaTex. You just need "[/tex]" at the end, not "[ tex ]".

The simplest way to do this would be to take the origin of your coordinate system to be at the vertex so that the equation is just [itex]y= ax^2[/itex]. That is, the parabola passes through (0,0), (3/2, 2), and (-3/2, 2) and you determine a by [itex]2= a(3/2)^2[/itex].

From that, the "full" volume is [itex]\int_{-3/2}^{3/2}2- ax^2 dx[/itex] and the depth half full, h, is given by [itex]a\int_{-3/2}^{3/2} h- ax^2 dx= (1/2)\int_{-3/2}^{3/2}2- ax^2 dx[/itex].
 
  • #6
I agree with Halls of Ivy that you should probably look at the way you're integrating.
Once you've gotten the proper equation for a parabola, I see two methods. .
A) Make a rectangle from the x-axis up to your line y = 2, ending at the points (+1.5, 2) and (-1.5, 2). What is the area of this box?
How does the area given by the definite integral relate to this?
B) More tricky method: Take the inverse of the parabola, and integrate directly the area of water, and use that.
 
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1. What is the "Exciting riverbed problem"?

The "Exciting riverbed problem" refers to a phenomenon observed in rivers where the riverbed becomes unstable and begins to move or shift, causing potential hazards for nearby structures and ecosystems.

2. What causes the "Exciting riverbed problem"?

The exact cause of the "Exciting riverbed problem" is still being studied, but it is believed to be a combination of factors such as changes in water flow, sediment movement, and erosion.

3. What are the potential consequences of the "Exciting riverbed problem"?

The "Exciting riverbed problem" can have serious consequences for both human structures and natural ecosystems. It can lead to erosion of river banks, damage to bridges and other structures, and disruption of aquatic habitats.

4. How can the "Exciting riverbed problem" be prevented?

Preventing the "Exciting riverbed problem" involves careful monitoring and management of river systems. This can include strategies such as building protective structures, controlling water flow, and implementing erosion control measures.

5. Is the "Exciting riverbed problem" a common occurrence?

The "Exciting riverbed problem" can occur in any river system, but it is more common in areas where there is significant human development or changes in land use. It is also more likely to occur during periods of heavy rainfall or flooding.

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