Question

The angles of elevation of the top of a tower from two points at a distance of 5 meters and 20 meters along the same straight line from the base of the tower, are complementary. Find the height of the tower.

• A. 10 m
• B. 15 m
• C. \(10\sqrt{3}\) m
• D. 20 m

Hint:

There is a direct formula for this specific problem: If the angles of elevation of the top of a tower from two points at distances 'a' and 'b' from its base are complementary, then the height of the tower 'h' is given by \( h = \sqrt{ab} \). Here, \(h = \sqrt{5 \times 20} = \sqrt{100} = 10\) m.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a trigonometry problem involving angles of elevation. Complementary angles are two angles that add up to 90 degrees. We can use the tangent function to relate the height of the tower to the distances and angles.
Step 2: Key Formula or Approach:
Let 'h' be the height of the tower.
Let the two points be at distances 'a' and 'b' from the base (a=5m, b=20m).
Let the angles of elevation be \( \theta \) and \( 90^{\circ} - \theta \). A standard result for this setup is that the height of the tower \(h = \sqrt{ab}\).
Step 3: Detailed Explanation:
Let the height of the tower be h. The two points are at distances 5 m and 20 m from the base.
Let the angle of elevation from the point at 20 m be \( \theta \).
So, \( \tan(\theta) = \frac{h}{20} \) -- (1)
The angle of elevation from the point at 5 m is complementary, so it is \( 90^{\circ} - \theta \).
So, \( \tan(90^{\circ} - \theta) = \frac{h}{5} \) -- (2)
Using the trigonometric identity \( \tan(90^{\circ} - \theta) = \cot(\theta) \), we can rewrite equation (2):
\( \cot(\theta) = \frac{h}{5} \)
Since \( \cot(\theta) = \frac{1}{\tan(\theta)} \), we can multiply the two tangent expressions:
\( \tan(\theta) \times \cot(\theta) = \left(\frac{h}{20}\right) \times \left(\frac{h}{5}\right) \)
We know that \( \tan(\theta) \times \cot(\theta) = 1 \).
\[ 1 = \frac{h^2}{100} \]
\[ h^2 = 100 \]
\[ h = \sqrt{100} = 10 \] m (Height must be positive).
Step 4: Final Answer:
The height of the tower is 10 m.