Question 1 of 12intermediate🔧 ApplyNumerical3 marks

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Correct Answer

The height of the tree is 83\sqrt{3} m.

Exercise: EXERCISE 9.1 | Q: 2 | (Page Page 9)
For More Understanding

Explanation

This is a classic application of trigonometry from Exercise 9.1, Question 2. The problem involves a right triangle formed when the tree breaks. The broken part becomes the hypotenuse, and we use tan 30° and cos 30° to find the vertical height and the length of the broken part respectively.

Solution Steps

  1. Step 1: Let the tree break at point B. The foot of tree is A, and the top touches ground at C. Given: AC = 8 m and angle between broken part and ground = 30°.

  2. Step 2: In right triangle ABC, angle at C = 30°. Using tan 30° = AB/AC, we get 1/3\sqrt{3} = AB/8. Therefore, AB = 8/3\sqrt{3} m (unbroken standing part).

  3. Step 3: Using cos 30° = AC/BC, we get 3\sqrt{3}/2 = 8/BC. Therefore, BC = 16/3\sqrt{3} m (broken part).

  4. Step 4: Total height of tree = AB + BC = 8/3\sqrt{3} + 16/3\sqrt{3} = 24/3\sqrt{3} = 83\sqrt{3} m.

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