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 can use trigonometric ratios. Here’s how we can find the distance of the foot of the ladder from the wall:

### Step 1: Understand the triangle formed
We have a right triangle formed by the wall, the ground, and the ladder. The ladder acts as the hypotenuse, the wall is the opposite side, and the distance from the wall to the foot of the ladder is the adjacent side.

### Step 2: Identify the given values
- Length of the ladder (hypotenuse) = 10 meters
- Angle of inclination (θ) = 30 degrees

### Step 3: Use the cosine function
We can use the cosine of the angle to find the distance from the wall (adjacent side). The cosine function is defined as:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]

Substituting the known values:
\[ \cos(30^\circ) = \frac{\text{Base}}{10} \]

### Step 4: Find the cosine of 30 degrees
From trigonometric tables or knowledge, we know:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

### Step 5: Set up the equation
Now we can substitute this value into the equation:
\[ \frac{\sqrt{3}}{2} = \frac{\text{Base}}{10} \]

### Step 6: Solve for the Base
To find the base (distance from the wall), we can rearrange the equation:
\[ \text{Base} = 10 \cdot \frac{\sqrt{3}}{2} \]
\[ \text{Base} = 5\sqrt{3} \]

### Step 7: Substitute the value of \(\sqrt{3}\)
Given that \(\sqrt{3} = 1.732\), we can substitute this value into the equation:
\[ \text{Base} = 5 \cdot 1.732 \]

### Step 8: Calculate the distance
Now we perform the multiplication:
\[ \text{Base} = 5 \cdot 1.732 = 8.66 \text{ meters} \]

Thus, the distance of the foot of the ladder from the wall is **8.66 meters**.

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