Shadow of doubt

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 5 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?

The figure below can represent the problem:


[ a ]

To solve this, the variables must be reflected into the diagram.


[ b ]



[ c ]

Figure 1. Illustration of the problem.

The movement of the shadow at any given time is represented by dy/dt. The problem is, which diagram is correct? Are these the same?

From Figure 1b, the basic equation becomes:



The derivative is:



On the other hand, from Figure 1c, the equation becomes:



By differentiation,



Now, it is obvious that the two figures are different because the ensuing equations are not the same.

Figure 1b is the change of the shadow from the point 40 ft from the post. On the other hand, Figure 1c is the movement of the shadow from the post.

Which one is correct?

Taking another look, Figure 1b presents the rate of change of the shadow initially from 40 ft from the post but this reference point apparently moves as the man walks. It is not a fixed point. On the other hand, the post as a common fixed reference point both for the man and his shadow is presented in Figure 1c. However, it is interesting to note also that there is no shadow behind the man or at the length x. Is it right to use Figure 1b where the shadow is confined in front of the man?

This could have been easier to resolve if the question asks: How fast is the tip of his shadow moving from the post when he is 40 ft from it? The original sentence in the problem essentially casts a shadow of doubt.

Only the final answer was given in the book which implies Figure 1c must be correct.

It is also interesting to note that the problem can be solved even without using the point 40 ft from the post. Therefore, the movement of the shadow is constant or the same wherever the man is.

The point 40 ft can be useful if the solution took the longer route of using the quotient rule of differentiation from the basic equation:



The derivative becomes:



where y should be solved first from similar triangles.

In all cases,

The answer is .

I checked the book again. Calculus: Early Transcendentals Volume 1 by Stewart.

His first name?

James.

Ok, I believe him.


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Reference
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