Question

A tall tower has its base at point K. Three points A, B and C are located at distances of 4 metres, 8 metres and 16 metres respectively from K. The angles of elevation of the top of the tower from A and C are complementary. What is the angle of elevation (in degrees) of the tower’s top from B?

• A. 30
• B. We need more information to solve this.
• C. 60
• D. 45
• E. 15

Answer 1

The Correct Option is D

Given:

• Point A is 4 meters from K.
• Point B is 8 meters from K.
• Point C is 16 meters from K.

The angles of elevation from points A and C are complementary. Complementary angles add up to \(90^\circ\).

To find the angle of elevation from point B, consider the following steps:

1. Let the height of the tower be \(h\).
2. Using trigonometric properties, for point A:
   \[ \tan(\theta_A) = \frac{h}{4} \]
3. For point C, let the angle of elevation be \((90^\circ - \theta_A)\) since they are complementary:
   \[ \tan(90^\circ - \theta_A) = \frac{h}{16} \]
4. The trigonometric identity for complementary angles is \(\tan(90^\circ - \theta) = \cot(\theta)\), thus:
   \[ \cot(\theta_A) = \frac{h}{16} \]
5. The reciprocal relationship between tangent and cotangent gives:
   \[ \cot(\theta_A) = \frac{1}{\tan(\theta_A)} \]
6. Equating the expressions for \(\tan(\theta_A)\) and \(\cot(\theta_A)\):
   \[ \frac{h}{4} = \frac{16}{h} \]
7. Cross-multiplying results in:
   \[ h^2 = 64 \]
8. Solving for \(h\), we find:
   \[ h = 8 \]
9. To find the angle of elevation from point B (8 meters from K):
   \[ \tan(\theta_B) = \frac{h}{8} = \frac{8}{8} = 1 \]
10. The angle \(\theta_B\) where \(\tan(\theta_B) = 1\) is:
    \[ \theta_B = 45^\circ \]

Therefore, the angle of elevation of the tower's top from B is 45 degrees.