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.
42 m
Idea: At the same time of day, the Sun’s rays make the same angle with the ground for all objects. So the pole and the tower form two right triangles that are similar (same angle of elevation of the Sun).
For the pole: height = 6 m, shadow = 4 m.
For the tower: height = \(h\) m (unknown), shadow = 28 m.
Each pair “object + shadow” forms a right triangle with the ground.
Because the angle of elevation of the Sun is the same,
\[\triangle \text{(pole)} \sim \triangle \text{(tower)}.\]
Heights correspond to heights, shadows to shadows:
\[\frac{\text{height of pole}}{\text{shadow of pole}} = \frac{\text{height of tower}}{\text{shadow of tower}}\]
\[\frac{6}{4} = \frac{h}{28}.\]
Cross-multiply:
\[6 \times 28 = 4h\]
\[168 = 4h\]
\[h = \frac{168}{4} = 42.\]
Conclusion: The height of the tower is 42 m.