A 10 metre long ladder is placed against a wall. It is inclined at an angle of 30° to the ground. The distance (in m) of the foot of the ladder from the wall is (Given `sqrt(3)` = 1.732)

Options

A. `8.16`
B. `7.32`
C. `8.26`
D. `8.66`

Answer

To solve the problem step by step, we will use the properties of a right triangle formed by the ladder, the wall, and the ground.

### Step 1: Understand the Triangle
We have a right triangle where:
- The ladder forms the hypotenuse (PA) with a length of 10 meters.
- The angle between the ladder and the ground is 30°.
- The distance from the wall to the foot of the ladder (PB) is the base of the triangle.

### Step 2: Use Trigonometric Ratios
In a right triangle, we can use the cosine function to find the base (PB):
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Here, \(\theta = 30^\circ\), the adjacent side is PB, and the hypotenuse is PA (10 m).

### Step 3: Calculate PB
Using the cosine of 30°:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
So, we can write:
\[ \cos(30^\circ) = \frac{PB}{10} \]
Substituting the value of \(\cos(30^\circ)\):
\[ \frac{\sqrt{3}}{2} = \frac{PB}{10} \]
Now, solve for PB:
\[ PB = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \]

### Step 4: Substitute the Value of \(\sqrt{3}\)
Given that \(\sqrt{3} \approx 1.732\):
\[ PB = 5 \times 1.732 = 8.66 \text{ meters} \]

### Step 5: Conclusion
The distance of the foot of the ladder from the wall is approximately 8.66 meters.

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