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

To solve this problem, we utilize the concept of trigonometry, specifically the sine function. The given ladder acts as the hypotenuse of the right triangle formed by the ladder, the wall, and the ground.

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side (in this case, the height of the wall) to the length of the hypotenuse (the ladder). Therefore, we have:

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

Inserting the given values:

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

We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, so we substitute:

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

To solve for \(h\), the height of the wall, multiply both sides by 14:

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

Calculating further:

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

However, we attempted to solve it initially with a misconception. Hence, resolving correctly with the correct formula and steps, we find:

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

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

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

Therefore, the height of the wall is indeed 7 m.