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Math Help - cross sections

  1. #1
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    cross sections

    Question is
    Let f(x, y) = 100 + 10x + 25y x^2 5y^2.
    a. Describe the cross section of the surface Z = f(x, y) produced by cutting it with the planes Y = 0, y = 1, y = 2, and y = 3.

    b. Describe the cross section of the surface in the planes x = 0, x = 1, x = 2, and x = 3.
    c. Describe the surface z = f(x, y)


    What I have done for part b:
    Z = F(0,y) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(0) + 25y (0)^2 5y^2
    = 100 + 25y - 5y^2

    z = F(1,y) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(1) + 25y (1)^2 5y^2
    = 111 + 25y - 5y^2

    z = F(2,y) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(2) + 25y (2)^2 5y^2
    = 124 + 25y - 5y^2

    z = F(3,y) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(3) + 25y (3)^2 5y^2
    = 139 + 25y - 5y^2

    How i described the results:
    The cross section of f(x, y) = 100 + 10x + 25y x2 5y2 that correspond to x = 0, x = 1, x = 2, and x = 3 are upper semicircles with centres (0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0) respectively.

    Is this correct so far?
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  2. #2
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    Quote Originally Posted by mellowdano View Post
    Question is
    Let f(x, y) = 100 + 10x + 25y x^2 5y^2.
    a. Describe the cross section of the surface Z = f(x, y) produced by cutting it with the planes Y = 0, y = 1, y = 2, and y = 3.
    b. Describe the cross section of the surface in the planes x = 0, x = 1, x = 2, and x = 3.
    c. Describe the surface z = f(x, y)

    What I have done for part b:
    Z = F(0,y) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(0) + 25y (0)^2 5y^2
    = 100 + 25y - 5y^2
    Which is a parabola opening downward.

    CB
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  3. #3
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    For part A i did the following and corrcted my explanation
    Z = F(x,0) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10x + 25(0) x^2 5(0)^2
    = -x^2 + 10x + 100z

    F(x,1) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10x + 25(1) x^2 5(1)^2
    = - x^2 + 10x + 120

    z = F(x,2) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10x + 25(2) x^2 5(2)^2
    = - x^2 + 10x + 130

    z = F(x,3) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10x + 25(3) x^2 5(3)^2
    = - x^2 + 10x + 130

    The cross section of f(x, y) = 100 + 10x + 25y x^2 5y^2 produced by cutting it with the plane y = 0, y = 1, y = 2, and y = 3 shows that each of these is a parabola that opens downward.

    What I have done so far for part C
    Describe the surface z = f(x, y)
    z = f(0, 0) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(0) + 25(0) (0)^2 5(0)^2
    = 100

    z = f(1, 1) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(1) + 25(1) (1)^2 5(1)^2
    = 141

    z = f(2, 2) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(2) + 25(2) (2)^2 5(2)^2
    = 194

    z = f(3, 3) = 100 + 10x + 25y x^2 5y^2
    = 100 + 10(3) + 25(3) (3)^2 5(3)^2
    = 259

    My textbook has not really explained to me what I had to do for part C so I assumed I did the above correctly. I am not sure how to explain my results.
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