A railroad tunnel is shaped like a semiellipse as shown below.
The height of the tunnel at the center is 58 ft and the vertical clearance must be 29 ft at a point 21 ft from the center. Find an equation for the ellipse.

- anonymous

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- anonymous

Diagram: http://prntscr.com/7mjbvf

- jim_thompson5910

not sure, but let me think

- anonymous

Anything?

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## More answers

- jim_thompson5910

is there anything about the ellipse not mentioned? I feel like there's some missing info

- anonymous

nope

- anonymous

vertex is 0,58 and there are two points on the ellipse at +-21,29

- jim_thompson5910

if the ellipse is taller than it is wide, then a = 58 and this pairs up with the y^2 term

- jim_thompson5910

and if (21,29) is on the ellipse, then (x,y) = (21,29) --> x = 21 and y = 29

- jim_thompson5910

assume the center is (h,k) = (0,0)

- jim_thompson5910

\[\Large \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2} = 1\]
\[\Large \frac{(21-0)^2}{b^2}+\frac{(29-0)^2}{58^2} = 1\]
solve for b

- anonymous

24.25ish

- jim_thompson5910

actually you don't have to solve for b
you can stop at b^2

- anonymous

well f***

- jim_thompson5910

since b^2 is in the denominator under the x^2 term

- anonymous

I'm getting a "Max Iterations Error" on my calculator

- jim_thompson5910

what kind of calculator do you have?

- anonymous

TI-36x Pro

- jim_thompson5910

I'm not familiar with that type

- jim_thompson5910

but you should get b^2 = 588

- anonymous

It has a number solve function where I can type in 2 sides of an equation and It'll solve for a variable in the equation. That's what I was using and it gave that error. When I solved for b^2 it works

- anonymous

ok

- jim_thompson5910

hmm strange

- anonymous

ok so we found b^2 and b, so now what?

- jim_thompson5910

replace b^2 with 588 and a^2 with whatever 58^2 is

- jim_thompson5910

(h,k) = (0,0)

- jim_thompson5910

x and y are left alone in the equation

- anonymous

so \[\large \frac{ x^2 }{ 588 }+\frac{ y^2 }{ 3364 } =1\]

- anonymous

then we need to add the values

- anonymous

@jim_thompson5910

- jim_thompson5910

add values?

- anonymous

is that the equation or is there something left?

- jim_thompson5910

the last thing you posted is the equation they want

- jim_thompson5910

I guess you could solve for y to get some expression in the form y = ...
that will get you the top half of the ellipse, which is the tunnel

- anonymous

should I do that?

- jim_thompson5910

hmm now I'm not sure if they want the full ellipse or just the upper half ellipse

- anonymous

ill give both

- jim_thompson5910

good idea

- anonymous

So do I just solve for y=?

- jim_thompson5910

yeah

- anonymous

So I'm stuck at \[\frac{ y^2 }{ 3364 }=1-\frac{ x^2 }{ 588 }\]

- jim_thompson5910

multiply both sides by 3364
then take the square root of both sides
you focus on the positive square root because you want the upper half

- anonymous

how I multiply x^2/588 by 3364 and what would that come out to

- jim_thompson5910

\[\Large \frac{ y^2 }{ 3364 }=1-\frac{ x^2 }{ 588 }\]
\[\Large 3364*\frac{ y^2 }{ 3364 }=3364*(1-\frac{ x^2 }{ 588 })\]
\[\Large y^2=3364*(1-\frac{ x^2 }{ 588 })\]
\[\Large y^2=3364*1-3364*\frac{ x^2 }{ 588 }\]
\[\Large y^2=3364-\frac{3364x^2 }{ 588 }\]
\[\Large y^2=3364-\frac{841x^2 }{ 147 }\]
\[\Large y=???\]

- anonymous

It was that last part I was confused about.\[y=58-\frac{ 29x }{ 7\sqrt{3} }\]

- jim_thompson5910

you can't take the square root like that

- anonymous

well crap

- jim_thompson5910

\[\Large y^2=3364-\frac{841x^2 }{ 147 }\]
\[\Large \sqrt{y^2}=\sqrt{3364-\frac{841x^2 }{ 147 }}\]
\[\Large y=\sqrt{3364-\frac{841x^2 }{ 147 }}\]

- jim_thompson5910

you apply the square root to the entire side

- anonymous

then what? Is that all we can do

- jim_thompson5910

yeah that's as far as you can go

- anonymous

Great thanks

- jim_thompson5910

\[\Large \sqrt{x + y} \ne \sqrt{x} + \sqrt{y}\]

- anonymous

It's been a long day of math for me. Finally off to bed. Thanks for all the help, dude!

- jim_thompson5910

no problem

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