a man is walking at the rate of 1.5 m/s toward a street light which is 5m above the level ground. at what rate is the tip of his shadow moving if the man is 2m tall

To solve this problem, we'll use related rates from calculus.

Let's denote:
- \( h \) as the height of the street light (5 meters).
- \( y \) as the height of the man (2 meters).
- \( x \) as the distance between the man and the base of the street light.
- \( s \) as the length of the man's shadow.

Given:
- The man is walking toward the street light at a rate of \(\frac{dx}{dt} = -1.5\) m/s (negative because \(x\) is decreasing).

We need to find \(\frac{ds}{dt}\), the rate at which the tip of the shadow is moving.

From similar triangles, we have:
\[ \frac{h}{x + s} = \frac{y}{s} \]

Substituting the given values:
\[ \frac{5}{x + s} = \frac{2}{s} \]

Cross-multiplying gives:
\[ 5s = 2(x + s) \]

Expanding and simplifying:
\[ 5s = 2x + 2s \]
\[ 3s = 2x \]
\[ s = \frac{2x}{3} \]

Now, differentiate both sides with respect to time \( t \):
\[ \frac{d}{dt}\left( s \right) = \frac{d}{dt}\left( \frac{2x}{3} \right) \]

Using the chain rule:
\[ \frac{ds}{dt} = \frac{2}{3} \cdot \frac{dx}{dt} \]

Substituting \(\frac{dx}{dt} = -1.5\) m/s:
\[ \frac{ds}{dt} = \frac{2}{3} \cdot (-1.5) \]
\[ \frac{ds}{dt} = -1 \text{ m/s} \]

Thus, the rate at which the tip of his shadow is moving is \(-1\) m/s. This means the tip of his shadow is moving towards the street light at a rate of 1 m/s.

Would you like further details or have any questions about this solution?

Here are 5 questions you might ask next:
1. Can you explain related rates in more detail?
2. How do you apply the chain rule in related rates problems?
3. Can you solve another related rates problem involving different shapes?
4. How does the rate of change of the shadow's length relate to the man's walking speed?
5. What are some common mistakes to avoid in related rates problems?

**Tip:** Always draw a diagram to visualize the problem when dealing with related rates, as it helps in understanding the relationships between different quantities.

Explore a related rates problem involving a man walking towards a street light. Understand how to apply calculus concepts like related rates and the chain rule to find the rate of change of his shadow's tip. Suitable for advanced high school students studying calculus.

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