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.

Question

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 · Answer

@Loser66

anonymous · Answer

attachment

anonymous · Answer

@campbell_st

danjs · Answer

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 · Answer

plug in values for x and y?

danjs · Answer

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

danjs · Answer

for the b value

anonymous · Answer

(0,54) is good right?

danjs · Answer

yes, and (8,18)

anonymous · Answer

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

danjs · Answer

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

anonymous · Answer

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

danjs · Answer

it will cross the y axis at + and - b

danjs · Answer

multiply by b^2, then take square root

danjs · Answer

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

anonymous · Answer

oh lol, which beans I get +-54?

anonymous · Answer

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

danjs · Answer

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

danjs · Answer

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

danjs · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

danjs · Answer

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

danjs · Answer

[1/3]^2 = 1/9

anonymous · Answer

is my final equation correct?

anonymous · Answer

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

danjs · Answer

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

anonymous · Answer

so how do I change the final equation?

danjs · Answer

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

danjs · Answer

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

anonymous · Answer

ok thanks

danjs · Answer

welcome