anonymous one year ago 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.

1. anonymous

@Loser66

2. anonymous

3. anonymous

@campbell_st

4. 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....

5. anonymous

plug in values for x and y?

6. DanJS

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

7. DanJS

for the b value

8. anonymous

(0,54) is good right?

9. DanJS

yes, and (8,18)

10. anonymous

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

11. DanJS

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

12. anonymous

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

13. DanJS

it will cross the y axis at + and - b

14. DanJS

multiply by b^2, then take square root

15. DanJS

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

16. anonymous

oh lol, which beans I get +-54?

17. anonymous

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

18. DanJS

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

19. DanJS

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

20. DanJS

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

21. anonymous

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

22. anonymous

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

23. anonymous

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

24. DanJS

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

25. DanJS

[1/3]^2 = 1/9

26. anonymous

is my final equation correct?

27. anonymous

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

28. DanJS

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

29. anonymous

so how do I change the final equation?

30. DanJS

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

31. DanJS

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

32. anonymous

ok thanks

33. DanJS

welcome