The angle of elevation of a ladder leaning against a wall is `60^@` and the foot of the ladder is 4.6 metre away from the wall . The length of the ladder is

Options

A. 2.3 metre
B. 4.6 metre
C. 9.2 metre
D. 7.8 metre

Answer

To find the length of the ladder leaning against the wall, we can use trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the problem
We have a right triangle formed by the wall, the ground, and the ladder. The angle of elevation of the ladder is given as \(60^\circ\), and the distance from the wall to the foot of the ladder (the base of the triangle) is \(4.6\) meters.

### Step 2: Identify the sides of the triangle
- Let \(AC\) be the length of the ladder (hypotenuse).
- Let \(AB\) be the height of the wall (opposite side).
- Let \(BC\) be the distance from the wall to the foot of the ladder (adjacent side), which is \(4.6\) meters.

### Step 3: Use the cosine function
From the right triangle, we can use the cosine of the angle to relate the adjacent side and the hypotenuse:
\[ \cos(60^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AC} \]
Substituting the known values:
\[ \cos(60^\circ) = \frac{4.6}{AC} \]

### Step 4: Substitute the value of \(\cos(60^\circ)\)
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]
So we can write:
\[ \frac{1}{2} = \frac{4.6}{AC} \]

### Step 5: Solve for \(AC\)
Cross-multiplying gives us:
\[ AC = 4.6 \times 2 \]
Calculating this gives:
\[ AC = 9.2 \text{ meters} \]

### Conclusion
The length of the ladder is \(9.2\) meters.

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