This problem involves the concept of similar triangles. The relationship between the height of a pole and the length of its shadow remains constant at a given time of day because the angle of the sunlight is the same for both poles.

Given:

• Height of the first pole = 3 m
• Shadow length of the first pole = 3.6 m
• Shadow length of the second pole = 54 m
• Height of the second pole = x (to be found)

Since the two poles and their shadows form similar right-angled triangles, their corresponding sides are proportional:

$\frac{\text{Height of first pole}}{\text{Shadow of first pole}} = \frac{\text{Height of second pole}}{\text{Shadow of second pole}}$

Substituting the given values:

$\frac{3}{3.6} = \frac{x}{54}$

Solving for $x$:

1. Compute $\frac{3}{3.6}$:

   $\frac{3}{3.6} = \frac{30}{36} = \frac{5}{6}$

2. Set up the proportion:

   $\frac{5}{6} = \frac{x}{54}$

3. Cross multiply:

   $5 \times 54 = 6 \times x$

   $270 = 6x$

4. Solve for $x$:

   $x = \frac{270}{6} = 45$

Final Answer:

The height of the second pole is 45 meters.

Would you like a further breakdown or explanation?

Related Questions:

1. How does the concept of similar triangles apply to real-life problems?
2. What happens if the shadow length changes at different times of the day?
3. How can we use trigonometry to find the angle of elevation of the sun?
4. Can we generalize this method for any number of poles and shadows?
5. How would the calculation change if the ground were not level?

Tip:

Always check if two triangles are similar before using proportions. In shadow problems, the key factor is that the sun's rays create the same angle for all objects at the same time.