For any heights and distances problem where the angles of elevation from two points at distances \(a\) and \(b\) from the base of a tower are complementary, the height of the tower is always given by:
\[ h = \sqrt{ab} \]
Here, \(a = 7\) and \(b = 16\), so \(h = \sqrt{7 \times 16} = 4\sqrt{7}\text{ m}\) instantly.
Step 1: Understanding the Question:
This question is from Trigonometry, specifically Heights and Distances.
We are given a vertical tower of height \(h\) with two points on the ground, C and B, aligned with the base D.
The distances are \(CD = 7\text{ m}\) and \(CB = 9\text{ m}\), making the total distance \(BD = 7 + 9 = 16\text{ m}\).
The angles of elevation from C and B are complementary, i.e., \(\theta\) and \(90^{\circ} - \theta\). We need to determine \(h\).
Step 2: Key Formula or Approach:
We use the trigonometric definition of tangent:
\[ \tan \phi = \frac{\text{Perpendicular}}{\text{Base}} \]
We also use the complementary angle trigonometric identity:
\[ \tan(90^{\circ} - \theta) = \cot\theta \]
By setting up two equations for \(\tan\theta\) and \(\cot\theta\), we can eliminate the angle by multiplying them, since \(\tan\theta \cdot \cot\theta = 1\).
Step 3: Detailed Explanation:
From the right-angled triangle \(\triangle ACD\):
The angle of elevation at C is \(\theta\).
The base \(CD = 7\text{ m}\).
\[ \tan\theta = \frac{h}{7} \quad \text{(Equation 1)} \]
From the right-angled triangle \(\triangle ABD\):
The angle of elevation at B is \(90^{\circ} - \theta\).
The base \(BD = CD + CB = 7 + 9 = 16\text{ m}\).
\[ \tan(90^{\circ} - \theta) = \frac{h}{16} \]
Using the identity \(\tan(90^{\circ} - \theta) = \cot\theta\):
\[ \cot\theta = \frac{h}{16} \quad \text{(Equation 2)} \]
Multiply Equation 1 and Equation 2:
\[ \tan\theta \times \cot\theta = \frac{h}{7} \times \frac{h}{16} \]
Since \(\tan\theta \times \cot\theta = 1\):
\[ 1 = \frac{h^2}{112} \]
\[ h^2 = 112 \]
Take the square root of both sides:
\[ h = \sqrt{112} \]
Simplify the radical:
\[ h = \sqrt{16 \times 7} = 4\sqrt{7}\text{ m} \]
Step 4: Final Answer:
The height 'h' is \(4\sqrt{7}\) m.
