a vertical pole of length 15m casts a shadow 10m.At the same time a flag pole casts a shadow of 60 m long on the ground.Find height of the pole.

To solve this problem, we can use the concept of similar triangles. Since both the pole and the flagpole cast shadows at the same time, the angles of elevation of the sun to the tops of the pole and the flagpole are the same. This implies that the triangles formed by the poles and their shadows are similar.

Given:
- Height of the pole: 15 meters
- Length of the pole's shadow: 10 meters
- Length of the flagpole's shadow: 60 meters
- Height of the flagpole: \( h \) meters (unknown)

We can set up the proportion based on the similarity of the triangles:

\[
\frac{\text{Height of pole}}{\text{Length of pole's shadow}} = \frac{\text{Height of flagpole}}{\text{Length of flagpole's shadow}}
\]

Substituting the given values:

\[
\frac{15}{10} = \frac{h}{60}
\]

Now, solve for \( h \):

\[
h = \frac{15}{10} \times 60 = 1.5 \times 60 = 90 \text{ meters}
\]

So, the height of the flagpole is 90 meters.

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

### Relative Questions
1. What are similar triangles, and how are they useful in solving real-world problems?
2. How can you determine the height of a tree using similar triangles and its shadow?
3. What other methods can be used to calculate the height of a tall object?
4. How would the problem change if the sun's position changed?
5. Can this method be used to find the height of a building? If so, how?
6. What is the angle of elevation, and how does it relate to shadows?
7. How would you solve this problem if only the angle of elevation was given?
8. Why does the proportion method work in solving problems like these?

### Tip
Always check if the triangles involved in a problem are similar by comparing their angles or by using trigonometric ratios.

Learn how to calculate the height of a vertical pole using the concept of similar triangles. This problem involves determining the height of a pole based on its shadow length and comparing it to a flagpole's shadow length. Suitable for Grades 7-9.

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