A railroad tunnel is shaped like a semiellipse, as shown below.

The height of the tunnel at the center is 54 ft, and the vertical clearance must be 18 ft at a point 8 ft from the center. Find an equation for the ellipse.

Question

A railroad tunnel is shaped like a semiellipse, as shown below.

The height of the tunnel at the center is 54 ft, and the vertical clearance must be 18 ft at a point 8 ft from the center. Find an equation for the ellipse.

alexh107 · Answer

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mortonsalt · Answer

Let's recall the form of a semi-ellipses: 
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
Wherein we let the centre be (0,0).

mortonsalt · Answer

This ellipse will then go through (0,54), (0,-54) and (8, 18).

mortonsalt · Answer

Plug the values in to find a and b, then calculate a.

mortonsalt · Answer

This will then give you the equation of the ellipse.

alexh107 · Answer

What would the value of b be? @mortonsalt

mortonsalt · Answer

If you plug in (0,54) to the equation, you'll be able to solve it. :)

mortonsalt · Answer

Since x^2/a^2 will then be a 0.

mortonsalt · Answer

Leaving you with one variable to solve for.

alexh107 · Answer

0/a^2 + 54^2/b^2 = 1 is what I have so far

alexh107 · Answer

Oh okay, one sec I'll try to solve it

mortonsalt · Answer

Yup! You're on the right track. Once you've solved for b, you'd be able to solve for a by plugging in (8,18). I'm not entirely sure what your final solution is (I've yet to solve it myself).

alexh107 · Answer

Does b = 54?

alexh107 · Answer

@mortonsalt

mortonsalt · Answer

Should be. :) I'll let you handle it from here on out.