Solving a Conics Question: Bridge Arch Height at Point 30m from Center

[msimard8]

Hello,

I started this problem. I don't know really how to set it up. I attached my work. I know it is wrong but I do not know where. the correct answer is 42.2.

Here is the question:

A bridge over a river is supported by a hyperbolic arch which is 200 m wide at the base. The maximum height of the arch is 50 m. How high is the arch at a point 30 m from the center.

I drew a diagram (which I know is incorrect because my work assumes the center is (0,0)

Help.

If you have any ideas without looking at my work, anything will be appreciated. Thanks

 

[quantumdude]

Hi, sorry for the late response. Blame it on the summer!

Anyway let's take a look at the standard form of the equation of a hyperbola with a vertical transverse axis.

$$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$$

You've drawn one of the vertices at the origin, which is fine. But then you set $(h,k)=(0,0)$ which is not fine. Those are the coordinates of the center, which certainly does not coincide with either of the vertices. You've also misidentified $a$ and $b$. They are not the distances given in the problem.

Here's what I would do. Start from the diagram that you've drawn (with the vertex at the origin). That means that the center of the hyperbola is on the y-axis, which implies that $h=0$ in the above equation. Then use the 3 points on your diagram to find $a$, $b$, and $k$. You have 3 points and you need to find 3 constants. That should be feasible.