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Let's solve this problem using related rates.

### Problem Restatement
We have:
- A street light mounted on a \( 15 \)-foot pole.
- A man who is \( 6 \) feet tall walking away from the pole at \( 4 \) ft/s.
- We need to find how fast the tip of his shadow is moving when he is \( 30 \) feet from the pole.

### Diagram and Variables
Let's denote:
- \( x \): the distance between the man and the pole (in feet).
- \( y \): the length of the man’s shadow (in feet).

The rate of change of \( x \) with respect to time \( t \) is given by:
\[
\frac{dx}{dt} = 4 \, \text{ft/s}
\]

### Using Similar Triangles
The triangles formed by the pole and the tip of the shadow, and by the man and his shadow are similar. Therefore, we can write the following proportion based on their corresponding sides:

\[
\frac{15}{x + y} = \frac{6}{y}
\]

Solving for \( y \) in terms of \( x \):

1. Cross-multiply:
   \[
   15y = 6(x + y)
   \]
2. Expand and solve for \( y \):
   \[
   15y = 6x + 6y
   \]
   \[
   9y = 6x
   \]
   \[
   y = \frac{2}{3}x
   \]

### Differentiate with Respect to Time
To find how fast the tip of the shadow is moving, we need \( \frac{d}{dt}(x + y) \):

1. Substitute \( y = \frac{2}{3}x \):
   \[
   x + y = x + \frac{2}{3}x = \frac{5}{3}x
   \]

2. Differentiate \( x + y \) with respect to \( t \):
   \[
   \frac{d}{dt}(x + y) = \frac{5}{3} \cdot \frac{dx}{dt}
   \]

3. Substitute \( \frac{dx}{dt} = 4 \):
   \[
   \frac{d}{dt}(x + y) = \frac{5}{3} \cdot 4 = \frac{20}{3} \approx 6.67 \, \text{ft/s}
   \]

### Answer
The tip of the man’s shadow is moving at approximately \( 6.67 \, \text{ft/s} \) when he is \( 30 \) feet from the pole.

Would you like further details or have any questions?

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Here are some related questions to consider:

1. How would the solution change if the man’s height was different?
2. What if the pole height was changed to a different value?
3. How does the speed of the man affect the rate at which the shadow moves?
4. What if the man was walking toward the pole instead of away?
5. How would this change if the ground was sloped rather than flat?

**Tip:** When dealing with related rates problems, always identify and relate all variables before differentiating to simplify the process.

A street light is mounted at the top of a 15-ft-tall pole. A man 6 feet tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast (in ft/s) is the tip of his shadow moving when he is 30 feet from the pole?

Solve a related rates problem where a 6-ft tall man walks away from a 15-ft pole, determining how fast the tip of his shadow moves. This question involves the concept of similar triangles and differentiation. Suitable for Grades 11-12.

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