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hawkfalcon

  • 2 years ago

The height of triangle ACE is 12 and base AE = 20. A rectangle is inscribed in triangle ACE as shown below.? a) Express the rectangle’s area as a function of x and y. Use the fact that the triangle ACE is similar to triangle BCD to write y as a function of x. c)Express the rectangle’s area as a function of only x. d) Find the maximum possible area of the rectangle.

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  1. hawkfalcon
    • 2 years ago
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    |dw:1345995812545:dw|

  2. hawkfalcon
    • 2 years ago
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    Nobody knows what to do?:#

  3. hawkfalcon
    • 2 years ago
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    @Zekarias ?

  4. Zekarias
    • 2 years ago
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    a) (12-y)x b) 12(x-0.1x^2) c) 30 sq units

  5. hawkfalcon
    • 2 years ago
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    ...How?

  6. Zekarias
    • 2 years ago
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    Is "How" for all of it?

  7. hawkfalcon
    • 2 years ago
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    Just b and c

  8. hawkfalcon
    • 2 years ago
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    especially b... :O

  9. Zekarias
    • 2 years ago
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    consider the perpendicular line form C, we can come out with the ratio 12/y =10/x, where 10 is half the rectangles side,20. Then simplify it so as to inset the result in to the equation in 'a'. To 'c' just use the application of derivative.

  10. hawkfalcon
    • 2 years ago
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    Okay. Makes sense I guess:p thanks!

  11. ganeshie8
    • 2 years ago
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    small catch

  12. ganeshie8
    • 2 years ago
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    triangle ACE is similar to triangle BCD => height1/height2 = base1/base2

  13. ganeshie8
    • 2 years ago
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    12/y = 20/x y = (3/5)x

  14. ganeshie8
    • 2 years ago
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    this would give you max area 60

  15. hawkfalcon
    • 2 years ago
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    o.O But y needs to be written as a function of x.

  16. ganeshie8
    • 2 years ago
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    y = (3/5)x

  17. ganeshie8
    • 2 years ago
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    its a function of x

  18. ganeshie8
    • 2 years ago
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    linear

  19. hawkfalcon
    • 2 years ago
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    Oh yeah :3. So why did you use the entire base and @Zekarias used 1/2?

  20. ganeshie8
    • 2 years ago
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    since both are similar triangles, "ratio of bases" equals "ratio of heights"

  21. hawkfalcon
    • 2 years ago
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    Okay, that makes sense. :3 Can you explain how you got 60 though?

  22. ganeshie8
    • 2 years ago
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    write the area equation in x and optimize

  23. ganeshie8
    • 2 years ago
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    for optimizing, use either of the below two : 1) its a parabola facing down so max value = f(-b/2a) or 2) use calculus

  24. ganeshie8
    • 2 years ago
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    this q from conics, or calculus ?

  25. hawkfalcon
    • 2 years ago
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    calculus. Going into calculus next year and they gave us some prep stuff*.*

  26. ganeshie8
    • 2 years ago
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    oh ok, so we have this so far : A = x(12-y) -------(1) y = (3/5)x --------(2)

  27. hawkfalcon
    • 2 years ago
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    Okay, true.

  28. ganeshie8
    • 2 years ago
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    hmm which one moves ?

  29. ganeshie8
    • 2 years ago
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    ohhk yeah as we go down its moving a lot it seems...

  30. ganeshie8
    • 2 years ago
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    substitute (2) in (1)

  31. hawkfalcon
    • 2 years ago
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    x(12-(3/5)x)

  32. ganeshie8
    • 2 years ago
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    \(A = x(12 - \frac{3}{5}x)\) \(A = 12x- \frac{3}{5}x^2)\)

  33. hawkfalcon
    • 2 years ago
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    Okayy and solve for x

  34. ganeshie8
    • 2 years ago
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    nope we need to optimize A

  35. ganeshie8
    • 2 years ago
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    find maximum possible value of A, right ?

  36. hawkfalcon
    • 2 years ago
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    Oh.

  37. hawkfalcon
    • 2 years ago
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    Ya

  38. ganeshie8
    • 2 years ago
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    theres a process for finding maximum/minimum values of any funcitons

  39. ganeshie8
    • 2 years ago
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    you familiar with differentiation ?

  40. hawkfalcon
    • 2 years ago
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    ... I guess I havenet learned that yet..

  41. ganeshie8
    • 2 years ago
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    its simple... if you knw differentiation already

  42. hawkfalcon
    • 2 years ago
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    I think we are learning that next year :( (monday)

  43. ganeshie8
    • 2 years ago
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    just if you knw derivative of x, and x^2, its enough

  44. ganeshie8
    • 2 years ago
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    oh ok then you cannot use calculus, lets switch to conics. you knw parabolas, right

  45. hawkfalcon
    • 2 years ago
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    Okay, I don't know how to do that.. Maybe I should make a new question asking for the dirivative of A=12x-3/4x^2

  46. hawkfalcon
    • 2 years ago
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    Yes.

  47. hawkfalcon
    • 2 years ago
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    You said f(-b/2a)

  48. ganeshie8
    • 2 years ago
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    yes

  49. ganeshie8
    • 2 years ago
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    A = 12x - (3/5) x^2

  50. ganeshie8
    • 2 years ago
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    \(A = \frac{-3}{5}x^2 + 12x\)

  51. hawkfalcon
    • 2 years ago
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    Yes

  52. ganeshie8
    • 2 years ago
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    a = -3/5 b = 12

  53. hawkfalcon
    • 2 years ago
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    Cool, thanks, just plug and chug now:D

  54. ganeshie8
    • 2 years ago
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    f(-b/2a) gives the vertex y co-ordinate, which is the maximum value

  55. hawkfalcon
    • 2 years ago
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    y=10

  56. ganeshie8
    • 2 years ago
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    |dw:1345999525420:dw|

  57. ganeshie8
    • 2 years ago
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    -b/2a = 10 f(-b/2a) = ?

  58. ganeshie8
    • 2 years ago
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    put x = -b/2a = 10, in area function

  59. hawkfalcon
    • 2 years ago
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    Okay.

  60. hawkfalcon
    • 2 years ago
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    60!:D

  61. ganeshie8
    • 2 years ago
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    if you remember, for any parabola, -b/2a is the x-coordinate of vertex f(-b/2a) is the y-coordinate of vertex

  62. ganeshie8
    • 2 years ago
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    great !!

  63. hawkfalcon
    • 2 years ago
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    Thank you so much! Now I can do the rest of these problems:D

  64. ganeshie8
    • 2 years ago
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    glad to hear ! calculus wud be much fun :) good luck !!

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