A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.
Step-by-Step Solution
Key Concept: The sun’s rays are parallel, therefore the triangles formed by the pole and its shadow and by the tower and its shadow are similar. Hence the corresponding sides are in the same ratio.
1. Draw the two right‑angled triangles\
- Triangle \(\triangle POB\) for the pole: \(PO = 4\,\text{m}\) (shadow) and \(OB = 6\,\text{m}\) (height).\
- Triangle \(\triangle T O'B'\) for the tower: \(O'T' = 28\,\text{m}\) (shadow) and \(O'B' = h\) (height to be found).\
Both triangles share the same angle of elevation of the sun, so they are similar.
2. Set up the proportion using corresponding sides\
$$\frac{\text{height of pole}}{\text{shadow of pole}} = \frac{\text{height of tower}}{\text{shadow of tower}}$$\
$$\frac{6}{4} = \frac{h}{28}$$
3. Solve for \(h\)\
$$h = \frac{6}{4}\times 28 = \frac{3}{2}\times 28 = 42\ \text{metres}$$
4. State the answer\
The height of the tower is 42 m.
Correct Answer: 42 m