A vertical pole, 30 m long, casts a shadow 20 m long on the ground At the same time , a vertical builging casts a shadow 60 m lng on the ground. What is the height of the building ?

Options

A. 60 m
B. 120 m
C. 150 m
D. 90 m

Answer

To solve the problem, we will use the concept of similar triangles.

### Step-by-Step Solution:

1. **Identify the given information:**
- Height of the vertical pole (AB) = 30 m
- Length of the shadow of the pole = 20 m
- Length of the shadow of the building (QR) = 60 m
- We need to find the height of the building (PQ).

2. **Draw the triangles:**
- Draw a vertical line representing the pole (AB) and label its height as 30 m.
- Draw a horizontal line representing the shadow of the pole (BC) and label its length as 20 m.
- Draw a vertical line representing the building (PQ) and label its height as unknown.
- Draw a horizontal line representing the shadow of the building (QR) and label its length as 60 m.

3. **Establish the similarity of triangles:**
- The triangles formed by the pole and its shadow (triangle ABC) and the building and its shadow (triangle PQR) are similar because they share the same angle of elevation of the sun.
- Therefore, we can set up a proportion based on the heights and shadow lengths of the two triangles.

4. **Set up the proportion:**
\[ \frac{AB}{BC} = \frac{PQ}{QR} \]
Substituting the known values:
\[ \frac{30}{20} = \frac{PQ}{60} \]

5. **Cross-multiply to solve for PQ:**
\[ 30 \times 60 = 20 \times PQ \]
\[ 1800 = 20 \times PQ \]

6. **Divide both sides by 20 to find PQ:**
\[ PQ = \frac{1800}{20} = 90 \text{ m} \]

7. **Conclusion:**
- The height of the building (PQ) is 90 m.

### Final Answer:
The height of the building is **90 m**.