The angle of elevation of a ladder leaning against a wall is $60^\circ$ and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is :

### Concept & Cosine Ratio When a ladder leans against a wall, it forms the hypotenuse of a right-angled triangle. The distance from the wall is the adjacent side. We use the cosine function: $$ \cos(\theta) = \frac{\text{Adjacent (Distance from wall)}}{\text{Hypotenuse (Length of ladder)}} $$ ### Step-by-Step Solution 1. Given: Angle of elevation ($\theta$) = $60^\circ$. Distance of foot from the wall (Adjacent) = $4.6$ m. 2. Let the length of the ladder be $L$. 3. Apply the cosine ratio: $$ \cos(60^\circ) = \frac{4.6}{L} $$ 4. Substitute the value of $\cos(60^\circ) = \frac{1}{2}$: $$ \frac{1}{2} = \frac{4.6}{L} $$ 5. Cross-multiply to solve for $L$: $$ L = 4.6 \times 2 = 9.2 \text{ m} $$ ### Exam Strategy & Shortcut In a right triangle with a $60^\circ$ angle at the base, the hypotenuse is exactly twice the length of the adjacent base. Simply double the given distance: $4.6 \times 2 = 9.2$. ### Common Pitfall Using tangent or sine instead of cosine, which would incorrectly yield the height of the wall on which the ladder is leaning, rather than the ladder's length itself. ### Final Answer Therefore, the correct answer is **9.2 m**.