A railroad tunnel is shaped like a semiellipse as shown below.
A semiellipse is shown on the coordinate plane with vertices on the x axis and one point of intersection with the positive y axis.
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.
I will post a pic in a second.

- anonymous

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- anonymous

@Loser66

- anonymous

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- anonymous

@campbell_st

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## More answers

- DanJS

hey gave 2 points, and it is centered at the origin.
x^2 / a^2 + y^2 / b^2 = 1
use the two points to find a and b....

- anonymous

plug in values for x and y?

- DanJS

yeah.. as a check , one of the points is already on the y axis

- DanJS

for the b value

- anonymous

(0,54) is good right?

- DanJS

yes, and (8,18)

- anonymous

0^2/a^2 so that cancels out and I have:
54^2/b^2 = 1

- DanJS

yeah it crosses the y axis at 54, so that checks out for b right

- anonymous

2916/b^2 = 1
how do I simplify?

- DanJS

it will cross the y axis at + and - b

- DanJS

multiply by b^2, then take square root

- DanJS

should just say 54^2 = b^2 , so b is 54

- anonymous

oh lol, which beans I get +-54?

- anonymous

ok and then I plug that back in with the point 8,18 right?

- DanJS

yep, they really gave you that value, but it checks correct

- DanJS

need to find + and - a, where it crosses x axis

- DanJS

use the other point, and the b value, solve a

- anonymous

8^2/a^2 + 18^2/54^2 = 1
64/a^2 + 324/2916= 1 how do I simplify

- anonymous

I got a = 6 * sqrt(2)
Is that right?

- anonymous

so I get:
x^2/72 + y^2/2916 = 1
for the final equation is that right?

- DanJS

when you have \[\frac{ 18^2 }{ 54^2 } = [\frac{ 18 }{ 54 }]^2\]
reduce the inner fraction first, then square it

- DanJS

[1/3]^2 = 1/9

- anonymous

is my final equation correct?

- anonymous

x^2/72 + y^2/2916 = 1?

- DanJS

yes that is right for a, but remember it is + and -

- anonymous

so how do I change the final equation?

- DanJS

for the standard form, you want to leave those denominators as a^2 and b^2, not expanded out

- DanJS

\[\frac{ x^2 }{ [6\sqrt{2}]^2 }+\frac{ y^2 }{ 54^2 } = 1\]

- anonymous

ok thanks

- DanJS

welcome

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