Thread: Quadratic Relations

Hi,
I would like to ask your help, I can't figure out how to do that question.
A concrete bridge over a river has an underside in the shape of a parabolic arch. At the water level, the arch is 30m wide. It has a maximum height of
10m above the water. The minimum vertical thickness of the concrete is 1.5m
a)Find algebric relation that represents the shape of the arch.
When I tries my answer was y=1.5(x-25)+10 was that right?
b)What is the vertical thickness of the concrete 3m from the centre of the arch?
c)If the water level rises 2m, how wide will the arch be at this new level?

Thank you very much!

Hello, Universe!

I'm not sure how the "thickness" works, so I'll skip part (b)

A concrete bridge over a river has an underside in the shape of a parabolic arch.
At the water level, the arch is 30m wide.
It has a maximum height of 10m above the water.
The minimum vertical thickness of the concrete is 1.5m

(a) Find an equation that represents the shape of the arch.
(b) What is the vertical thickness of the concrete 3m from the centre of the arch?
(c) If the water level rises 2m, how wide will the arch be at this new level?

Code:

                      10|
                       ***
                  *     |     *
              *         |         *
            *           |           *
           *            |            *
                        |
      ----*-------------+-------------*----
         -15            |            15

(a) The parabola opens downward and is symmetric to the y-axis.
. . Its equation is of the form: .$\displaystyle y \:=\:-ax^2 + c$

We know that $\displaystyle (0,10)$ is on the parabola.
. . We have: .$\displaystyle 10 \:=\:-a(0^2) + c\quad\Rightarrow\quad c = 10$
The equation (so far) is: .$\displaystyle y \:=\:-ax^2 + 10$

We know that $\displaystyle (15,0)$ is on the parabola.
. . We have: .$\displaystyle 0 \:=\:-a(15^2) + 10\quad\Rightarrow\quad a = \frac{2}{45}$

The equation is: .$\displaystyle \boxed{y \:=\:-\frac{2}{45}x^2 + 10}$


(c) If the water level rises 2 m, the bridge looks like this:

Code:

                      10|
                       ***
                  *     |     *
              *         |         *
            *           |           *
         --*-----------2+------------*--
                        |
      - - * - - - - - - + - - - - - - * - - 
                        |

When $\displaystyle y = 2$, we have: .$\displaystyle -\frac{2}{45}x^2 + 10 \:=\:2\quad\Rightarrow\quad x^2 = 180\quad\Rightarrow\quad x = \pm6\sqrt{5}$

The width of the bridge will be: .$\displaystyle 2 \times 6\sqrt{5}\:\approx\:\boxed{26.8\text{ meters}}$

The answer in the book says y=-17/450x^2+8.5 for part a.
Any ideas?
Thanks

Originally Posted by Universe

The answer in the book says y=-17/450x^2+8.5 for part a.
Any ideas?
Thanks

The wording of your problem is ambiguous, it seems to imply that all the data
given about the arch pertains to the inner surface. This is the assumption
that soroban has mad, and that I would have made.

However, given the book solution it is clear that the "height" is the height of
the outer edge of the arch, while the width is of the inner edge, so soroban's
diagram should look like:

Code:

                     8.5|
                       ***
                  *     |     *
              *         |         *
            *           |           *
           *            |            *
                        |
      ----*-------------+-------------*----
         -15            |            15

Which will give your books equation for the arch.

RonL

Thanks for the explantion I finally figured that out.