Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff. From a point on the ground 30 m away from the tower, wires are attached to the top and bottom of the flagstaff making angles of elevation \(60^\circ\) and \(30^\circ\) respectively. Find the height of the tower and lengths of the wires attached. (Take \(\sqrt{3} = 1.73\))

Hint:

Using \(\cos\theta\) directly to find the hypotenuse is much faster and more accurate than finding the perpendicular height first and then applying Pythagoras' theorem.

Answer 1

Step 1: Understanding the Question:
This is a height and distance trigonometry problem.
A flagstaff is mounted on top of a vertical tower.
From a point on the ground \(30\text{ m}\) away from the base of the tower, two wires are connected to the top and bottom of the flagstaff.
The angles of elevation to these two points are \(60^\circ\) and \(30^\circ\).
We need to calculate:
1. The height of the tower.
2. The lengths of the two wires.

Step 2: Key Formula or Approach:
1. Draw a clean geometric diagram representing the tower, flagstaff, ground point, and elevation angles.
2. Use standard trigonometric ratios in right-angled triangles:
\[ \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \]
\[ \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]

Step 3: Detailed Explanation:
1. Let \(CD\) represent the vertical tower of height \(h\).
2. Let \(AC\) represent the vertical flagstaff of height \(x\) surmounting the tower.
3. Let \(P\) be the point on the ground which is \(30\text{ m}\) away from the base \(D\). So, \(PD = 30\text{ m}\).
4. The wire attached to the bottom of the flagstaff is \(PC\), making an angle of elevation \(\angle CPD = 30^\circ\).
5. The wire attached to the top of the flagstaff is \(PA\), making an angle of elevation \(\angle APD = 60^\circ\).
6. Find the height of the tower (\(CD = h\)):
- In right-angled triangle \(\Delta PDC\):
\[ \tan 30^\circ = \frac{CD}{PD} \]
\[ \frac{1}{\sqrt{3}} = \frac{h}{30} \]
\[ h = \frac{30}{\sqrt{3}} = 10\sqrt{3}\text{ m} \]
- Using \(\sqrt{3} = 1.73\):
\[ h = 10 \times 1.73 = 17.3\text{ m} \]
7. Find the length of the wire attached to the bottom of the flagstaff (\(PC\)):
- In right-angled triangle \(\Delta PDC\):
\[ \cos 30^\circ = \frac{PD}{PC} \]
\[ \frac{\sqrt{3}}{2} = \frac{30}{PC} \]
\[ PC = \frac{60}{\sqrt{3}} = 20\sqrt{3}\text{ m} \]
- Using \(\sqrt{3} = 1.73\):
\[ PC = 20 \times 1.73 = 34.6\text{ m} \]
8. Find the length of the wire attached to the top of the flagstaff (\(PA\)):
- In right-angled triangle \(\Delta PDA\):
\[ \cos 60^\circ = \frac{PD}{PA} \]
\[ \frac{1}{2} = \frac{30}{PA} \]
\[ PA = 60\text{ m} \]

Step 4: Final Answer:
The height of the tower is \(17.3\text{ m}\).
The lengths of the wires attached to the top and bottom of the flagstaff are \(60\text{ m}\) and \(34.6\text{ m}\) respectively.