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: Problem Definition:
A ladder of 14 m length reaches the top of a vertical wall. The angle between the ladder and the wall is \(60^\circ\). Determine the wall's height.
This scenario forms a right-angled triangle with:
- Hypotenuse: Ladder length (\(h = 14\) m),
- Opposite side to the angle: Wall height,
- Angle between ladder and wall: \(60^\circ\).

Step 2: Trigonometric Calculation:
The sine function in a right-angled triangle is defined as:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] With \(\theta = 60^\circ\), the opposite side being the wall height (\(h_{\text{wall}}\)), and the hypotenuse being the ladder length (\(L = 14\) m), we get:
\[ \sin(60^\circ) = \frac{h_{\text{wall}}}{14} \] Given \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), the equation becomes:
\[ \frac{\sqrt{3}}{2} = \frac{h_{\text{wall}}}{14} \] Solving for \(h_{\text{wall}}\) by multiplying both sides by 14:
\[ h_{\text{wall}} = 14 \times \frac{\sqrt{3}}{2} = 7\sqrt{3} \]