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 m
• C. 14 m
• D. $7\sqrt{3}$ m

Answer 1

The Correct Option is B

The problem is solved using trigonometry, specifically the sine function. The ladder represents the hypotenuse of a right triangle formed by the ladder, wall, and ground.

The sine of an angle in a right triangle is the ratio of the opposite side (wall height) to the hypotenuse (ladder length). The formula is:

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$

Substituting the given values:

$$\sin(60^\circ) = \frac{h}{14}$$

Knowing that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, we get:

$$\frac{\sqrt{3}}{2} = \frac{h}{14}$$

To find \(h\), multiply both sides by 14:

$$h = 14 \cdot \frac{\sqrt{3}}{2}$$

This simplifies to:

$$h = \frac{14\sqrt{3}}{2} = 7\sqrt{3}$$

An initial misconception occurred. The correct approach involves:

$$h = 14 \cdot \cos(60^\circ)$$

Since $\cos(60^\circ) = \frac{1}{2}$:

$$h = 14 \cdot \frac{1}{2} = 7$$

Thus, the height of the wall is 7 m.