ASK YOUR TEACHER A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole?

Tip of the shadow is moving at the rate of .

Step-by-step explanation:

Refer to the attachment.

Let AB be the height of the street light. So, AB = 15 ft.  

Let DE be the height of the man. So, DE = 6 ft.

Let BE be the distance between the man and light pole. Assume BE = x  

Let BC be the distance between tip of shadow and light pole. Assume BC = y

Now height of the shadow of the man is EC which can be calculated as follows,

BC = BE + EC  

y = x + EC  

Subtracting x,  

y – x = EC.

So there are two triangles that is, triangle ABC and triangle DEC. Assume that both the triangles are similar.  

So applying similar triangle property,

Substituting the value,

Cross multiplying,

Using distributive property,

Subtracting 6y,

Adding 15x,

Dividing by 9,

Multiply and divide by 3,

Differentiating above equation with respect to t,

Applying constant multiple rule,

Given that man walks away from the pole with a speed of 4 ft/s that is,

Substituting the value,

Therefore, tip of the shadow is moving at the rate of .

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