微分應用題

Question

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.

Answer 1

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.