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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.

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  1. anonymous
    • one year ago
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    @Loser66

  2. anonymous
    • one year ago
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  3. anonymous
    • one year ago
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    @campbell_st

  4. DanJS
    • one year ago
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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....

  5. anonymous
    • one year ago
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    plug in values for x and y?

  6. DanJS
    • one year ago
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    yeah.. as a check , one of the points is already on the y axis

  7. DanJS
    • one year ago
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    for the b value

  8. anonymous
    • one year ago
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    (0,54) is good right?

  9. DanJS
    • one year ago
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    yes, and (8,18)

  10. anonymous
    • one year ago
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    0^2/a^2 so that cancels out and I have: 54^2/b^2 = 1

  11. DanJS
    • one year ago
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    yeah it crosses the y axis at 54, so that checks out for b right

  12. anonymous
    • one year ago
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    2916/b^2 = 1 how do I simplify?

  13. DanJS
    • one year ago
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    it will cross the y axis at + and - b

  14. DanJS
    • one year ago
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    multiply by b^2, then take square root

  15. DanJS
    • one year ago
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    should just say 54^2 = b^2 , so b is 54

  16. anonymous
    • one year ago
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    oh lol, which beans I get +-54?

  17. anonymous
    • one year ago
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    ok and then I plug that back in with the point 8,18 right?

  18. DanJS
    • one year ago
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    yep, they really gave you that value, but it checks correct

  19. DanJS
    • one year ago
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    need to find + and - a, where it crosses x axis

  20. DanJS
    • one year ago
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    use the other point, and the b value, solve a

  21. anonymous
    • one year ago
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    8^2/a^2 + 18^2/54^2 = 1 64/a^2 + 324/2916= 1 how do I simplify

  22. anonymous
    • one year ago
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    I got a = 6 * sqrt(2) Is that right?

  23. anonymous
    • one year ago
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    so I get: x^2/72 + y^2/2916 = 1 for the final equation is that right?

  24. DanJS
    • one year ago
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    when you have \[\frac{ 18^2 }{ 54^2 } = [\frac{ 18 }{ 54 }]^2\] reduce the inner fraction first, then square it

  25. DanJS
    • one year ago
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    [1/3]^2 = 1/9

  26. anonymous
    • one year ago
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    is my final equation correct?

  27. anonymous
    • one year ago
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    x^2/72 + y^2/2916 = 1?

  28. DanJS
    • one year ago
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    yes that is right for a, but remember it is + and -

  29. anonymous
    • one year ago
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    so how do I change the final equation?

  30. DanJS
    • one year ago
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    for the standard form, you want to leave those denominators as a^2 and b^2, not expanded out

  31. DanJS
    • one year ago
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    \[\frac{ x^2 }{ [6\sqrt{2}]^2 }+\frac{ y^2 }{ 54^2 } = 1\]

  32. anonymous
    • one year ago
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    ok thanks

  33. DanJS
    • one year ago
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    welcome

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