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A ladder 10 ft leans against a vertical wall. The upper end slips down the wall at 5 ft/sec. How fast is the ladder turning when it takes an angle of 20 degrees with the ground.

  • Calculus -

    let the top of the ladder by y ft up on the wall
    let the angle made at the foot be Ø

    sinØ = y/10
    10 sinØ = y
    10 cosØ dØ/dt = dy/dt

    given: when Ø = 20° , dy/dt = -5 ( y is decreasing)

    10 cos20° dØ/dt = -5
    dØ/dt = -1/(2cos20°) = -.532 radians/sec

    or angle is decreasing at .532 radians / sec
    which is appr. 30.5° / sec

    (notice the angle is changing very rapidly at that moment, since the ladder has almost fallen)

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