? 發問於 科學及數學數學 · 1 十年前

Applications of Differentiation `

A lighthouse L is built on a small island 3km from the nesrst point A of the straight shoreline .If the lighthouse lamp rotates at 2 revolutions per minute, how fast is the light spot P on the shoreline moving when it is 4km from A?

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  • Hunter
    Lv 6
    1 十年前
    最愛解答

    L

    |

    3km

    |

    ------------4km----------------

    A........................................P

    Ap = AL * tan (angle ALP)

    dAP/dt

    = d(AL * tan (angle ALP))/dt

    = AL * d (tan (angle ALP))/dt ..............(AL is a constant)

    = AL * [sec (angle ALP)]^2 * d(angle ALP)/dt.......(d(angle ALP)/dt is the rotational rate)

    Rotational rate

    = 2 revolution per min.

    = 2 * 2 * pi / 60

    =4*pi /60 (radian per second)

    [ sec (angle ALP)]^2 =[ tan(angle ALP)]^2 + 1=(4/3)^2 + 1 = 16/9 + 1 = 25/9

    The rate of speed

    = d(AP)/dt

    = 3 * 25/9 * 4 * pi / 60

    =1.745km/second.

    or

    = 104.7km/minute.

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