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.
The height of the tower is 42 m.
Explanation
This question is from Exercise 6.3 (Q.15) and applies the concept of similar triangles to find heights and distances through indirect measurement. The key insight is that when the sun's rays fall on vertical objects at the same time, the angles of elevation are equal, creating similar triangles. This principle is demonstrated in Example 7 of the textbook, where AA similarity criterion is applied because both objects are vertical (90° angle) and the sun's rays create the same angle with both objects.
Solution Steps
Step 1: Let the height of the tower be h metres.
Step 2: Since both the pole and tower are vertical to the ground, each makes an angle of 90° with the ground. Also, at the same time, the sun's rays make the same angle with both objects.
Step 3: Therefore, by AA similarity criterion, the triangle formed by the pole and its shadow is similar to the triangle formed by the tower and its shadow.
Step 4: For similar triangles, the ratio of corresponding sides is equal: Height of pole / Shadow of pole = Height of tower / Shadow of tower
Step 5: Substituting values: 6/4 = h/28
Step 6: Solving for h: h = ()/4 = 168/4 = 42 m