Question

From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be \(45^\circ\). The height (in metres) of the tower is :

• A. \(10\sqrt{3}\)
• B. \(30\sqrt{3}\)
• C. 60
• D. 30

Hint:

In any right-angled triangle, if one of the acute angles is \(45^\circ\), the triangle is isosceles because the other acute angle is also \(180^\circ - 90^\circ - 45^\circ = 45^\circ\).
Therefore, the perpendicular and the base are always equal in length.
Since the distance to the base is 60 m, the height must automatically be 60 m!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem is a application of trigonometry (heights and distances).
We have a vertical tower. A point on the ground is situated at a distance of 60 meters from its base.
The angle of elevation from this point to the top of the tower is \(45^\circ\). We need to determine the height of the tower.

Step 2: Key Formula or Approach:
1. Model the situation as a right-angled triangle where:
- The height of the tower represents the perpendicular (\(h\)).
- The distance from the observer to the base of the tower represents the base (\(b = 60\ \text{m}\)).
- The angle of elevation is \(\theta = 45^\circ\).
2. Use the tangent trigonometric ratio:
\[ \tan \theta = \frac{\text{Perpendicular}}{\text{Base}} \]

Step 3: Detailed Explanation:
1. Let \(AB\) represent the vertical tower of height \(h\) meters.
2. Let \(C\) be the observation point on the ground, such that the distance from the foot of the tower \(B\) is \(BC = 60\ \text{m}\).
3. The angle of elevation is \(\angle ACB = 45^\circ\).
4. In right-angled triangle \(\Delta ABC\) (right-angled at \(B\)):
\[ \tan(\angle ACB) = \frac{AB}{BC} \]
5. Substitute the known values into the formula:
\[ \tan 45^\circ = \frac{h}{60} \]
6. Since the standard trigonometric value of \(\tan 45^\circ\) is 1:
\[ 1 = \frac{h}{60} \implies h = 60\ \text{meters} \]
This gives the height of the vertical tower as 60 meters.

Step 4: Final Answer:
The height of the tower is 60 meters, which corresponds to option (C).