Question

A ladder 14 m long just reaches the top of a vertical wall. If the ladder makes an angle of $60^\circ$ with the wall, then the height of the wall is:

• A. $14\sqrt{3}$ m
• B. $7\sqrt{3}$ m
• C. 14 m
• D. 7 m

Answer 1

The Correct Option is B

Step 1: Understanding the problem:
We are given a ladder of length 14 m that just reaches the top of a vertical wall. The ladder makes an angle of \(60^\circ\) with the wall, and we need to find the height of the wall.
We can visualize the situation as a right-angled triangle where:
- The ladder is the hypotenuse (\(h = 14\) m),
- The height of the wall is the opposite side of the angle,
- The angle between the ladder and the wall is \(60^\circ\).

Step 2: Using trigonometry to find the height:
In a right-angled triangle, the sine of an angle is given by:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] Here, the angle \(\theta = 60^\circ\), the opposite side is the height of the wall \(h_{\text{wall}}\), and the hypotenuse is the length of the ladder (\(L = 14\) m).
So, we have:
\[ \sin(60^\circ) = \frac{h_{\text{wall}}}{14} \] We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), so:
\[ \frac{\sqrt{3}}{2} = \frac{h_{\text{wall}}}{14} \] Multiplying both sides by 14:
\[ h_{\text{wall}} = 14 \times \frac{\sqrt{3}}{2} = 7\sqrt{3} \]