The angle of elevation of a ladder leaning a gainst a house is `60^(@)` and the foot of the ladder is 6.5 metres from the house . The length of the ladder is

Options

A. `(13)/(sqrt(3))` metres
B. 13 metres
C. 15 metres
D. 3.25 metres

Answer

To solve the problem, we will use trigonometric ratios in a right triangle. Here’s the step-by-step solution:

### Step 1: Understand the Problem
We have a ladder leaning against a house, forming a right triangle with the ground and the wall of the house. The angle of elevation from the ground to the top of the ladder is \(60^\circ\), and the distance from the foot of the ladder to the house is \(6.5\) meters.

### Step 2: Identify the Triangle
In the right triangle:
- Let \(A\) be the point where the ladder touches the wall (the top of the ladder).
- Let \(B\) be the point where the foot of the ladder is on the ground.
- Let \(C\) be the point where the foot of the ladder is from the wall.

Here, \(AB\) is the length of the ladder (which we need to find), \(BC\) is the distance from the wall to the foot of the ladder (which is \(6.5\) meters), and the angle \(CAB\) is \(60^\circ\).

### Step 3: Use Trigonometric Ratios
To find the length of the ladder \(AB\), we can use the cosine of the angle \(CAB\):
\[ \cos(60^\circ) = \frac{BC}{AB} \]
Where:
- \(BC = 6.5\) meters
- \(\cos(60^\circ) = \frac{1}{2}\)

### Step 4: Set Up the Equation
Substituting the known values into the equation:
\[ \frac{1}{2} = \frac{6.5}{AB} \]

### Step 5: Solve for \(AB\)
Cross-multiplying gives:
\[ AB \cdot \frac{1}{2} = 6.5 \]
\[ AB = 6.5 \cdot 2 \]
\[ AB = 13 \text{ meters} \]

### Conclusion
The length of the ladder is \(13\) meters.

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