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.

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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.

Mathematics
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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....
plug in values for x and y?
yeah.. as a check , one of the points is already on the y axis
for the b value
(0,54) is good right?
yes, and (8,18)
0^2/a^2 so that cancels out and I have: 54^2/b^2 = 1
yeah it crosses the y axis at 54, so that checks out for b right
2916/b^2 = 1 how do I simplify?
it will cross the y axis at + and - b
multiply by b^2, then take square root
should just say 54^2 = b^2 , so b is 54
oh lol, which beans I get +-54?
ok and then I plug that back in with the point 8,18 right?
yep, they really gave you that value, but it checks correct
need to find + and - a, where it crosses x axis
use the other point, and the b value, solve a
8^2/a^2 + 18^2/54^2 = 1 64/a^2 + 324/2916= 1 how do I simplify
I got a = 6 * sqrt(2) Is that right?
so I get: x^2/72 + y^2/2916 = 1 for the final equation is that right?
when you have \[\frac{ 18^2 }{ 54^2 } = [\frac{ 18 }{ 54 }]^2\] reduce the inner fraction first, then square it
[1/3]^2 = 1/9
is my final equation correct?
x^2/72 + y^2/2916 = 1?
yes that is right for a, but remember it is + and -
so how do I change the final equation?
for the standard form, you want to leave those denominators as a^2 and b^2, not expanded out
\[\frac{ x^2 }{ [6\sqrt{2}]^2 }+\frac{ y^2 }{ 54^2 } = 1\]
ok thanks
welcome

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