What Is the Equation of a Parabola for a Bridge Needing Specific Dimensions?

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Homework Help Overview

The discussion revolves around finding the equation of a parabola that describes the arch of a bridge, given specific dimensions for height and width. The problem includes constraints related to the clearance needed for a tanker and the effects of potential flooding on this clearance.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss placing the vertex of the parabola at the origin and the implications of this choice on the equation. There are varying opinions on the most useful form of the parabola's equation, with some suggesting the vertex form while others consider the factored form. Questions arise regarding the values needed to satisfy the height and width requirements, as well as the conditions for the tanker clearance.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem setup and the mathematical relationships involved. Some have provided insights into how to derive the necessary parameters for the parabola, while others are questioning the assumptions made about the vertex location and the resulting equations.

Contextual Notes

Participants note that the problem must be solved under a time constraint, as it is due the following morning. There is also mention of the need for the bridge to accommodate a specific clearance for a tanker, which adds complexity to the problem.

SAC2009
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Homework Statement


1. It is a word problem I need done by tomorrow morning. so please help:
A new bridge is being constructed. The space between the support needs to be 1050 feet; the height at the center of the arch needs to be 350 feet. An empty tanker needs a 280 foot clearance to pass beneath it.
a) find the equation of a parabola with these characteristics. Place the vertex at the origin.
b) How wide is the channel that the tanker can pass through?
c) If the river were to flood and rise 10 feet, how would the clearance of the bridge be affected?

Homework Equations


The equation for a vertical parabola is y=a(x-h)squared + k



The Attempt at a Solution


I tried putting the vertex at (0,0) and plugging in (525,350) to get the a value. I don't know if this is anywhere close to what I was supposed to do.
 
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Personally I find the equation
y = A (x - p)(x - q)
more useful, when you know where the parabola has to be zero.
Do you see why?
 
SAC2009 said:

Homework Statement


1. It is a word problem I need done by tomorrow morning. so please help:
A new bridge is being constructed. The space between the support needs to be 1050 feet; the height at the center of the arch needs to be 350 feet. An empty tanker needs a 280 foot clearance to pass beneath it.
a) find the equation of a parabola with these characteristics. Place the vertex at the origin.
b) How wide is the channel that the tanker can pass through?
c) If the river were to flood and rise 10 feet, how would the clearance of the bridge be affected?

Homework Equations


The equation for a vertical parabola is y=a(x-h)squared + k
And if you Place the vertex at the origin, y(x)= ax2+ k. "The space between the support needs to be 1050 feet" so y(1050/2)= y(525)= 0 (NOT "526,350). "the height at the center of the arch needs to be 350 feet" so y(0)= 350. That is enough to find a and k.
In order for the tanker to pass through, the height must be at least 280 feet. What values of x make y= 280?


The Attempt at a Solution


I tried putting the vertex at (0,0) and plugging in (525,350) to get the a value. I don't know if this is anywhere close to what I was supposed to do.
 
I thought that if the vertex was at the origin (0,0) that the equation would be y=axsquared, because the vertex is (h,k). I keep getting an extremely small number for k, which wouldn't work because then the width at that point would be way too small.
 
sorry, i meant an extremely small number for a
 
You are right. I put the origin at (0, 350).

Now, either y= kx2 or y= -x2 depending on whether you want the parabola opening upward or downward. The second would better represent a support for a bridge but, mathematically, either way works.

If you want to use y= kx2 then the bases of the support are at (525, 350) and (-525, 350) and you would need, to allow the tanker to pass, to solve y= x2= 280.
 

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