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Math Help - integration and coordinate geometry

  1. #1
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    integration and coordinate geometry

    sorry i was going through some past papers and i need help on some questions again.

    1)
    given that dy/dx = ((3-square root x)^2)/square root x, x>0, and that y=2/3 at x=1
    Find y in terms of x


    2)
    the line l2 passes through the origin O and has gradient -2. the lines l2 and y=1/3x-7 intersect at the point P.
    a) calculate the coordinates of P

    given that y=1/3x-7 crosses the y-axis at the point C,
    b) calculate the exact area of triangeOCP.

    Thanks!
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  2. #2
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    Quote Originally Posted by devilicious
    sorry i was going through some past papers and i need help on some questions again.
    1)
    given that dy/dx = ((3-square root x)^2)/square root x, x>0, and that y=2/3 at x=1
    Find y in terms of x

    2)
    the line l2 passes through the origin O and has gradient -2. the lines l2 and y=1/3x-7 intersect at the point P.
    a) calculate the coordinates of P

    given that y=1/3x-7 crosses the y-axis at the point C,
    b) calculate the exact area of triangeOCP.

    Thanks!
    Hello,

    to 1.: Integrate dy/dx and you'll get:
    \int{\frac{(3-\sqrt{x})^2}{\sqrt{3}}}dx=\frac{2}{3} \cdot (\sqrt{x}-3)^3+c
    Plug in the values you know and solve for c. You'll get c=6. So the equation of your function reads:
    f(x)=\frac{2}{3} \cdot (\sqrt{x}-3)^3+6

    to 2.: Use the slope-point-formula and you'll get : l2: y=-2x. Plug in this term for y into the 2nd equation: -2x=1/3x-7. Solve for x. x = 3. Sub in this value in one of the two equations and you'll get y = -6. So the coordinates of P are (3, -6).

    to 3.: C has the coordinates (0, -7). Therefore the base of the triangle lies on the y-axis and has a length of 7. The height of the triangle is the x-value of point P. Now use the formula for the area of a triangle:
    A=\frac{1}{2} \cdot b \cdot h\ \Rightarrow \ A=\frac{1}{2} \cdot 7 \cdot 3 = \frac{21}{2}

    Greetings

    EB
    Last edited by earboth; April 2nd 2006 at 12:36 AM.
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  3. #3
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    Quote Originally Posted by devilicious
    sorry i was going through some past papers and i need help on some questions again.
    ...
    2)
    the line l2 passes through the origin O and has gradient -2. the lines l2 and y=1/3x-7 intersect at the point P.
    a) calculate the coordinates of P
    given that y=1/3x-7 crosses the y-axis at the point C,
    b) calculate the exact area of triangeOCP.

    Thanks!
    Hello,

    I've attached a diagram to illustrate my solution on problem 2.a) and 2.b)

    Greetings

    EB
    Attached Thumbnails Attached Thumbnails integration and coordinate geometry-dreick_fl.gif  
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