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Math Help - Need help with ff.

  1. #1
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    Need help with ff.

    Hello all!

    I'm not really good in math... recently, our prof announced we would be having a test in the coming week, and he gave us the ff. problems as review questions... trouble is, I'm having a hell of a time with it! Please help!

    1) f(g(x)) = x+5/x-7 and g(x)=2x+1/3; what is f(x), and find f(x+1/X-1)

    2) A circle is inside a rectangle, and it is tangent to each of the longer sides at the Midpoint. Area of rectangle - Area of circle = 126 sq. cm, and the sum of their perimeters is 112cm. What are the dimensiosn of the rectangle? Use pi=22/7.

    3) Find x and y so that: their sum x sum of their squares is 5500; their difference x difference of their squares is 352.

    4) A cable from a suspension bridge hands in the form of a parabola, thereby distributing the load horizontally. The distance between one tower and another is 150m; the cables are 22m above the roadway (highest pt.); and te lowest pt. above the roadway is 7m.

    What is the vertical distance to the cable from a point on the roadway, 15m from the tower?

    Thanks everyone!!!
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  2. #2
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    Quote Originally Posted by umbraculum View Post
    ...
    2) A circle is inside a rectangle, and it is tangent to each of the longer sides at the Midpoint. Area of rectangle - Area of circle = 126 sq. cm, and the sum of their perimeters is 112cm. What are the dimensiosn of the rectangle? Use pi=22/7....
    Hello,

    let l be the length of the rectangle
    and r be the radius of the circle then you get the equations according to your problem:

    A_{rectangle}-A_{circle}: l \cdot 2r-\pi \cdot r^2=126

    p_{rectangle}+p_{circle}: 2l+4r+2\pi r=112

    (Unfortunately the Latex isn't working anymore. So I repeat my equations in plain text)

    equ(1): A_{rectangle}-A_{circle}: l * 2r - π* r^2=126

    equ(2): p_{rectangle}+p_{circle}: 2l+4r+2πr=112

    Now calculate l from equ(1) using π = 22/7 : l = (126+22/7*r^2)/(2r)
    and put this term into equ(2) instead of l:

    2*(126+22/7*r^2)/(2r) +4r+2πr=112

    After a few transformation you'll get a quadratic equation:

    94r^2 - 784r + 882 = 0

    which has 2 solutions: r = 7 or r = 63/47

    Now plug in these values to calculate l. You get:
    l = 20 or l = 2308/47

    EB
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  3. #3
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    Quote Originally Posted by umbraculum View Post
    ...

    4) A cable from a suspension bridge hands in the form of a parabola, thereby distributing the load horizontally. The distance between one tower and another is 150m; the cables are 22m above the roadway (highest pt.); and te lowest pt. above the roadway is 7m.
    What is the vertical distance to the cable from a point on the roadway, 15m from the tower?...
    Hello,

    use a coordinate system. Place the origin in the middle between the two towers at road level. Then the parabola has the general equation:

    p(x) = a*x^2 + 7

    You know that the height of the cable in a distance of 75 m from the origin is 22 m. Plug in these values to calculate a:

    22 = a * (75)^2 + 7 <===> a = 15/5625 = 1/375

    The complete equation of the parabola is:

    p(x) = 1/375*x^2 + 7

    15 m from the tower means that the x-value is 60. Plug in this value into the equation of the parabola:

    p(60) = 1/375*(60)^2+7 = 16.6 m

    I've attached a diagram which shows the situation.

    EB
    Attached Thumbnails Attached Thumbnails Need help with ff.-parab_bruecke.jpg  
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  4. #4
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    Quote Originally Posted by umbraculum View Post
    ...Please help!

    1) f(g(x)) = x+5/x-7 and g(x)=2x+1/3; what is f(x), and find f(x+1/X-1)
    ...
    Hello,

    if f(x) is the term of a function then f^(-1)(x) is the term of the inverse function if it actually exists.
    You ought to know that f(f^(-1)(x)) = f^(-1(f(x)) = x

    1. step: Calculate g^(-1)(x) = 1/2*x-1/6

    2. f(g(g^(-1)(x))) = f(x)

    Therefore: f(x) = (1/2*x - 1/6 + 5)/(1/2*x - 1/6 - 7) = (1/2*x + 29/6)/(1/2*x - 43/6) = (x + 29/3)/(x - 43/3)

    Now that you know f(x) you can calculate f((x+1)/(x-1)):

    f((x+1)/(x-1)) = ((x+1)/(x-1) + 29/3)/((x+1)/(x-1) - 43/3) =
    (x+1 + 29/3*x - 29/3)/(x-1))/(x+1 - 43/3*x + 43/3)/(x-1)) =
    (32/3*x - 26/3)/(-40/3*x + 46/3) = (32x-26)/(-40x+46) = (16x-13)/(-20x+26)

    EB
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  5. #5
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    Hello,umbraculum!

    This will be hard to explain without LaTeX . . .


    1) f(g(x)) = (x + 5)/(x - 7) and g(x) = (2x + 1)/3.

    (a) Find f(x)

    (b) Find f([x+1]/[x-1])

    Assume that: . f(x) .= .(ax + b)/(cx + d)

    . . . . . . . . . . . . . . .a[(2x+1)/3] + b
    Then: . f(g(x)) . = . --------------------
    . . . . . . . . . . . . . . .c[(2x+1)/3] + d

    . . . . . . . . . . . . . . . . . . . . . . .2ax + a + 3b
    Multiply top and bottom by 3: . -----------------
    . . . . . . . . . . . . . . . . . . . . . . .2cx + c + 3d


    This is equal to (x + 5)/(x - 7)

    . . . . . . . . . . 2ax + (a + 3b) . . . x + 5
    so we have: . ------------------ .= .-------
    . . . . . . . . . . 2cx + (c + 3d) . . . .x - 7


    Equate coefficients: . 2a .= .1 . . a + 3b .= .5
    . . . . . . . . . . . . . - - 2c .= .1 . . c + 3d .= .-7

    . . and we have: . a = 1/2, . b = 3/2
    . . . . . . . . . . - - -c = 1/2, . c = -5/2


    . . . . . . . . . . . . (1/2)x + 3/2 . . . . x + 3
    Then: . f(x) . = . --------------- . = . -------
    . . . . . . . . . . . . (1/2)x - 5/2 . . . . .x - 5

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  6. #6
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    Hello again, umbraculum!

    I found a solution to #3 . . .
    . . but there must be a better way!


    3) Find x and y so that:
    their sum times the sum of their squares is 5500.
    their difference times the difference of their squares is 352.

    We have: . (x + y)(x + y) .= .5500 . . . . x + xy + xy + y .= .5500 . [1]
    . . . .And: . .(x - y)(x - y) . = . 352 . . . . . .x - xy - xy + y . = . 352 . [2]

    . . . Add [1] and [2]: . . 2x + 2y . = .5852 . . . . .x + y . = . 2926 . [3]
    Subtract [1] and [2]: . 2xy + 2xy .= .5148 . . . . xy + xy .= .2574 . [4]

    We have: . (x + y)(x - xy + y) .= .2926 . [5]
    . . . .and: . . . . - - xy(x + y) . . . = .2574 . [6]

    . . . . . . . . . . . . . .(x + y)(x - xy + y) . . . .2926 . . . . . . x - xy + y . . .113
    Divide [5] by [6]: . ------------------------ . = . ------ . . . . -------------- .= .-----
    . . . . . . . . . . . . . . . . .xy(x + y) . . . . . . . . .2574 . . . . . . . . x + y . . . . . 117

    This simplifies to the quadratic: .117x - 250xy + 117y .= .0

    . . which factors: .(13x - 9y)(9x - 13y) .= .0

    . . and has roots: . y .= .9x/13 and 13x/9


    Substitute y = 9x/13 into [3]: .x + (9x/13) .= .2926

    . . which gives us: .x = 13, y = 9


    Substitute y = 13x/9 and we get: .x = 9, y = 13

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  7. #7
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    Well, my solution to 3 isn't any shorter than Soroban's, but I feel it's a bit more straightforward.

    It starts the same way as Soroban's:
    (x + y)(x^2 + y^2) = 5500
    (x - y)(x^2 - y^2) = 352

    Expanding both we get:
    x^3 + x^2y + xy^2 + y^3 = 5500
    x^3 - x^2y - xy^2 + y^3 = 352

    Add both equations:
    2x^3 + 2y^3 = 5852 ==> x^3 + y^3 = 2926

    Subtract both equations:
    2x^2y + 2xy^2 = 5148 ==> x^2y + xy^2 = 2574

    Now we take a turn away from Soroban's solution.

    Note that
    (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 = (x^3 + y^3) + 3(x^2y + xy^2)

    So
    (x + y)^3 = 2926 + 3*2574 = 10648 ==> x + y = 22

    Thus
    x + y = 22 and y = 22 - x

    Now take a new look at your first condition:
    (x + y)(x^2 + y^2) = 5500

    (22)(x^2 + [22 - x]^2) = 5500

    (22)(2x^2 - 44x + 484) = 5500

    2x^2 - 44x + 484 = 250

    x^2 - 22x + 242 = 125

    x^2 - 22x + 117 = 0

    (x - 9)(x - 13) = 0

    Thus x = 9, 13 which gives y = 13, 9 respectively.

    -Dan
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  8. #8
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    Lovely work, Dan!

    I tried to find ways to square/cube various expressions
    . . as you did, but didn't get very far.

    Nice going!

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  9. #9
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    Quote Originally Posted by Soroban View Post
    Lovely work, Dan!

    I tried to find ways to square/cube various expressions
    . . as you did, but didn't get very far.

    Nice going!

    I'm blushing!

    -Dan
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