## anonymous one year ago 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.

1. anonymous

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

2. jim_thompson5910

not sure, but let me think

3. anonymous

Anything?

4. jim_thompson5910

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

5. anonymous

nope

6. anonymous

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

7. jim_thompson5910

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

8. jim_thompson5910

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

9. jim_thompson5910

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

10. 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

11. anonymous

24.25ish

12. jim_thompson5910

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

13. anonymous

well f***

14. jim_thompson5910

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

15. anonymous

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

16. jim_thompson5910

what kind of calculator do you have?

17. anonymous

TI-36x Pro

18. jim_thompson5910

I'm not familiar with that type

19. jim_thompson5910

but you should get b^2 = 588

20. 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

21. anonymous

ok

22. jim_thompson5910

hmm strange

23. anonymous

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

24. jim_thompson5910

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

25. jim_thompson5910

(h,k) = (0,0)

26. jim_thompson5910

x and y are left alone in the equation

27. anonymous

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

28. anonymous

then we need to add the values

29. anonymous

@jim_thompson5910

30. jim_thompson5910

31. anonymous

is that the equation or is there something left?

32. jim_thompson5910

the last thing you posted is the equation they want

33. 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

34. anonymous

should I do that?

35. jim_thompson5910

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

36. anonymous

ill give both

37. jim_thompson5910

good idea

38. anonymous

So do I just solve for y=?

39. jim_thompson5910

yeah

40. anonymous

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

41. 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

42. anonymous

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

43. jim_thompson5910

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

44. anonymous

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

45. jim_thompson5910

you can't take the square root like that

46. anonymous

well crap

47. jim_thompson5910

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

48. jim_thompson5910

you apply the square root to the entire side

49. anonymous

then what? Is that all we can do

50. jim_thompson5910

yeah that's as far as you can go

51. anonymous

Great thanks

52. jim_thompson5910

$\Large \sqrt{x + y} \ne \sqrt{x} + \sqrt{y}$

53. anonymous

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

54. jim_thompson5910

no problem