math-geometry

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The sum of the radii of two concentric circles is 40. A chord with length 36 of the larger circle is trisected by the smaller circle. The area of the smaller circle is k*pi. Find the value of k. Express your answer as an exact decimal.

Please show work. Thank you very much.

  • math-geometry -

    r + R = 40
    so
    r = 40 - R

    pi r^2 = k pi
    so
    k = r^2

    right triangles from center to middle of chord
    r^2 - 6^2 = R^2 - 18^2
    R^2 -r^2 = 324-36 = 288

    so we have two equations
    R = 40 - r
    R^2 -r^2 = 288
    then
    (40-r)^2 -r^2 = 288
    1600 -80 r +r^2-r^2 = 288
    80 r = 1312
    r = 16.4
    r^2 = 268.96
    but k = r^2 = 268.96

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