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Best_Mathematician

  • 2 years ago

Calculus challenge

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  1. Best_Mathematician
    • 2 years ago
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    The type of bread chosen for this special calculus toast isn't the square sandwich shape, but the kind that is curved across the top. Imagine that the toast is composed of the curved part sitting atop the rectangular portion. The equation of the curved part of the toast is x2/4 + y2 = 1, and it sits directly and perfectly on top of a rectangle of height 3 inches. a) What are the equations of the rectangular boundaries? b) Graph the toast boundaries, making certain to include screen shots of the boundary equations, Window settings, and the graph. c) How would you find the length of the curve

  2. Best_Mathematician
    • 2 years ago
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    ok

  3. Best_Mathematician
    • 2 years ago
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    then

  4. modphysnoob
    • 2 years ago
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    once you find two x values , they are your answer for a)

  5. Best_Mathematician
    • 2 years ago
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    is x two negative values

  6. modphysnoob
    • 2 years ago
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    is x squred?

  7. Best_Mathematician
    • 2 years ago
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    ya here is the equation again \[\frac{ x^2 }{ 4 } + y^2 = 1\]

  8. modphysnoob
    • 2 years ago
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    ahh, that's different, this is ellipse x^2/4 + (y-3)^2 = 1

  9. modphysnoob
    • 2 years ago
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    now plug in y=3, x^2/4=1 x^2=4 x=2,-2 that's your part a

  10. Best_Mathematician
    • 2 years ago
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    sweet ur r doing grt go ahead

  11. modphysnoob
    • 2 years ago
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    part b is graphing,

  12. dlipson1
    • 2 years ago
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    Here's the graph, done without benefit of the above

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  13. Best_Mathematician
    • 2 years ago
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    i wud just graph that right

  14. Best_Mathematician
    • 2 years ago
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    oh thx @dlipson1 can u go further

  15. dlipson1
    • 2 years ago
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    Uh oh, I think that's a line integral, I'd have to look that up. Do you know anything about Stochastic Optimization?

  16. Best_Mathematician
    • 2 years ago
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    hey hey hey never mind...i know how to find the length of the curve thanks..but i dont knw do i need to find length of whole curve or just the bread as in ur graph

  17. dlipson1
    • 2 years ago
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    Well, the rectangle is trivial (is the side along the x-axis included?), the top is just half the ellipse, that's the only real calculus (integration) you have to do.

  18. Best_Mathematician
    • 2 years ago
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    so from negative 2 to 2...right

  19. dlipson1
    • 2 years ago
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    Yeah, use the top half, y = sqrt(...), then (I just looked it up): Length = integral (sqrt(1+(y')^2))dy

  20. Best_Mathematician
    • 2 years ago
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    ya and what kinda graphing calculator r u using dude

  21. dlipson1
    • 2 years ago
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    http://www.padowan.dk/ and Google --> http://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx

  22. Best_Mathematician
    • 2 years ago
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    thx...getting my next question...i wud give u a lot of awards but unfortunately this site doesnt aloow lol

  23. dlipson1
    • 2 years ago
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    "Graph" (from padowan.dk) gives me the curve length of 4.882, then +3+3 +4 for the entire perimeter.

  24. Best_Mathematician
    • 2 years ago
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    sweet thanks

  25. Best_Mathematician
    • 2 years ago
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    hey @dlipson1 one more thing, how wud we find area on top of the toast

  26. dlipson1
    • 2 years ago
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    The rectangle + 2*integral (by symmetry) from 0 to 2 of y = sqrt(1-x^2/4)... hmm, do we need substitution here?

  27. Best_Mathematician
    • 2 years ago
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    do u need the derivative...i have it

  28. Best_Mathematician
    • 2 years ago
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    -x/(4y-12)....now wht to do

  29. dlipson1
    • 2 years ago
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    I'm not sure dy/dx helps. I set up the integral, thought about a trig. substitution, multiplied through by the 2 (as sqrt(4)) to get int(sqrt(4-x^2))dx, which I found here, but I think there must be an easier way (like polar coordinates): http://answers.yahoo.com/question/index?qid=20071231235113AAAAfuP

  30. dlipson1
    • 2 years ago
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    Here is almost the same problem: http://www.youtube.com/watch?v=PSlsj0IP8R8

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