Question

The angle of elevation of the top of a tower at a distance of 10 m from its base is 60\(^\circ\); then the height of the tower is

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

Hint:

Remember the SOH CAH TOA mnemonic to choose the correct trigonometric ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a problem of heights and distances, which is an application of trigonometry. We can model the situation with a right-angled triangle where the tower is the perpendicular side, the distance from the base is the base side, and the angle of elevation is the angle between the base and the hypotenuse.

Step 2: Key Formula or Approach:
In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\text{Height}}{\text{Base}} \]

Step 3: Detailed Explanation:
Let \(h\) be the height of the tower (opposite side).
The distance from the base is 10 m (adjacent side).
The angle of elevation, \(\theta\), is \(60^\circ\).
Using the tangent ratio:
\[ \tan 60^\circ = \frac{h}{10} \] We know the standard value \(\tan 60^\circ = \sqrt{3}\).
\[ \sqrt{3} = \frac{h}{10} \] To find \(h\), multiply both sides by 10:
\[ h = 10\sqrt{3} \text{ m} \]

Step 4: Final Answer:
The height of the tower is \(10\sqrt{3}\) m. This corresponds to option (B).