A ladder is leaning against a wall. It makes a 60° angle with the wall. If the distance between foot of ladder and wall is 5.5 meters, find the length of the ladder.

Options

A. 9.5m
B. 10m
C. 10.5m
D. 11m

Answer

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

### Step 1: Understand the Triangle
We have a right triangle formed by the ladder (hypotenuse), the wall (vertical side), and the ground (horizontal side). The angle between the ladder and the wall is 60°, and the distance from the foot of the ladder to the wall (the horizontal side) is 5.5 meters.

### Step 2: Identify the Sides
- Let:
- \( AC \) be the length of the ladder (hypotenuse).
- \( AB \) be the distance from the wall to the foot of the ladder (base) = 5.5 meters.
- \( BC \) be the height of the ladder against the wall (opposite side).

### Step 3: Use the Cosine Function
Since we know the angle and the adjacent side (base), we can use the cosine function:
\[ \cos(60°) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AB}{AC} \]
Substituting the known values:
\[ \cos(60°) = \frac{5.5}{AC} \]

### Step 4: Substitute the Value of Cosine
We know that:
\[ \cos(60°) = \frac{1}{2} \]
So we can rewrite the equation:
\[ \frac{1}{2} = \frac{5.5}{AC} \]

### Step 5: Solve for the Length of the Ladder
To find \( AC \), we can cross-multiply:
\[ 1 \cdot AC = 2 \cdot 5.5 \]
\[ AC = 11 \text{ meters} \]

### Conclusion
The length of the ladder is 11 meters.

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