An arch of a bridge is built in the shape of a half an ellipse. it has a span of 100ft. The height of the arch is 30ft from the center is 16ft. Find the height at the center.

Question

anonymous · Answer

Let's find a formula for the ellipse:
$$(x-50)^2+ay^2=2500$$seems to work. Find a by plugging x=30 or x=-30 in for x, 16 for y, and solving for a.

anonymous · Answer

Then, use that formula with x=50 and solve for y.

anonymous · Answer

so the height then would be 17.5ft?

anonymous · Answer

Sorry, you should have plugged in 20 or 80, I miswrote.

anonymous · Answer

20 or 80 for which variable?

anonymous · Answer

For x in the initial process to find a. I said 30 or -30 when that's for the entire term.

anonymous · Answer

how'd you get that? the 20 or 80, where'd it come from?

anonymous · Answer

Those are what you need to be 30 feet from the center.

anonymous · Answer

so I enter either 20 or 80 initially as x then. how do I know what to put in as x the second time as I find the height, y

anonymous · Answer

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anonymous · Answer

That will have a simpler formula: $$x^2+ay^2=2500$$
Just insert 30 or -30 for x the first time and find a using 16 for y, then insert x=0 and solve for y.

anonymous · Answer

you mean insert 20 or 80 in for x right?

anonymous · Answer

Not that formula, I simplified the situation by placing the origin in the middle of the river.

anonymous · Answer

Ah, alright

anonymous · Answer

so then it's 20 ft tall in the center

anonymous · Answer

Yup!

anonymous · Answer

Awesome thanks so much!