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May 發問於 科學及數學數學 · 8 年前

微分應用題

A boy 1.4m tall is walking to a street lamp 4.9m high at a rate of 0.7 m/s.

a) Suppose the boy is y m from the street lamp and the length of his shadow

is x m . Find the rate at which the length of his shadow is decreasing?

b) Find the rate at which the tip of his shadow is moving.

1 個解答

評分
  • ?
    Lv 7
    8 年前
    最愛解答

    Use this diagram for analysis.

    With notations in the figure, define s m as the length between the tip of the shadow and the street lamp.

    The question states that dy/dt = -0.7.

    圖片參考:http://imgcld.yimg.com/8/n/HA00430218/o/2013081815...

    A boy 1.4 m tall is walking to a street lamp 4.9 m high at a rate of 0.7 m/s.

    a) Suppose the boy is y m from the street lamp and the length of his shadow is x m . Find the rate at which the length of his shadow is decreasing?

    The question is asking for dx/dt.

    By similar triangles,

    x/1.4 = (x+y)/4.9

    4.9x = 1.4(x+y) = 1.4x + 1.4y

    3.5x = 1.4y

    x = 0.4y

    dx/dt = 0.4 dy/dt = 0.4 (-0.7) = -0.28

    Therefore, the length of his shadow is decreasing at a rate of 0.28 m/s.

    b) Find the rate at which the tip of his shadow is moving.

    The question is asking for |ds/dt|.

    By similar triangles,

    s/4.9 = x/1.4

    1.4s = 4.9x

    s = 3.5x

    ds/dt = 3.5 dx/dt = 3.5 (-0.28) = -0.98

    That means, tip of his shadow is moving towards the street lamp at a rate of 0.98 m/s.

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