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We are given a problem where a man of height 1.7 meters walks away from a 5-meter lamppost at a speed of 1.3 m/s. We are asked to find the rate at which his shadow length is increasing.

### Step-by-step Solution:

Let:
- \( h_1 = 5 \, \text{meters} \) (height of the lamppost),
- \( h_2 = 1.7 \, \text{meters} \) (height of the man),
- \( x \) be the distance of the man from the lamppost,
- \( y \) be the length of the shadow,
- \( \frac{dx}{dt} = 1.3 \, \text{m/s} \) (rate at which the man is walking away from the lamppost),
- \( \frac{dy}{dt} \) is the rate we need to find (rate at which the shadow is lengthening).

Using similar triangles, we can relate the heights and the distances. From the geometry of the problem, the ratio of the height of the lamppost to the total distance from the lamppost (man + shadow) is the same as the height of the man to the length of his shadow:

\[
\frac{5}{x + y} = \frac{1.7}{y}
\]

Now solve for \( y \):

1. Cross-multiply to get:
\[
5y = 1.7(x + y)
\]

2. Expand:
\[
5y = 1.7x + 1.7y
\]

3. Group like terms:
\[
5y - 1.7y = 1.7x
\]
\[
3.3y = 1.7x
\]

4. Solve for \( y \):
\[
y = \frac{1.7x}{3.3}
\]

Now, differentiate both sides with respect to time \( t \):

\[
\frac{dy}{dt} = \frac{1.7}{3.3} \cdot \frac{dx}{dt}
\]

Substitute \( \frac{dx}{dt} = 1.3 \, \text{m/s} \):

\[
\frac{dy}{dt} = \frac{1.7}{3.3} \cdot 1.3
\]

\[
\frac{dy}{dt} = \frac{2.21}{3.3} \approx 0.670 \, \text{m/s}
\]

Thus, the rate at which the man's shadow is increasing in length is approximately **0.670 m/s**.

Would you like more details on this solution or have any questions? 

Here are five related questions to further your understanding:
1. What happens to the rate of shadow growth if the man walks faster?
2. How would the problem change if the lamppost was shorter?
3. Can we derive an expression for the total rate at which the tip of the shadow is moving?
4. How would the rate of change differ if the man's height changed?
5. What other real-world applications use similar triangle relationships?

**Tip:** Always keep track of units when solving related rates problems, as they help ensure your solution makes sense physically.

A man of height 1.7 meters walks away from a 5-meter lamppost at a speed of 1.3 m/s. Find the rate at which his shadow is increasing in length.

Solve a related rates problem where a man walks away from a 5-meter lamppost at 1.3 m/s. Learn how to calculate the rate at which his shadow grows using geometry and differentiation. Step-by-step solution provided.

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