A ladder leaning against a window of a house makes an angle of `60^(@)` with the ground. Ifthe distance of the foot of the ladder from the wall is 4.2 m, then the height of the point, where the ladder touches the window from the ground is Closest to:

Options

A. `7. 3 m`
B. ` 6.8 m`
C. ` 7.8 m`
D. ` 7 m`

Answer

To solve the problem step by step, we can use trigonometric relationships.

### Step 1: Understand the problem
We have a ladder leaning against a wall, forming an angle of \(60^\circ\) with the ground. The distance from the foot of the ladder to the wall is given as \(4.2\) meters. We need to find the height \(H\) at which the ladder touches the wall.

### Step 2: Identify the trigonometric relationship
In this scenario, we can use the tangent function, which relates the angle of elevation to the opposite side (height of the ladder on the wall) and the adjacent side (distance from the wall).

The formula for tangent is:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
where:
- \(\theta = 60^\circ\)
- opposite = height \(H\)
- adjacent = distance from the wall = \(4.2\) m

### Step 3: Substitute the values into the formula
We can write:
\[ \tan(60^\circ) = \frac{H}{4.2} \]

### Step 4: Calculate \(\tan(60^\circ)\)
We know that:
\[ \tan(60^\circ) = \sqrt{3} \approx 1.732 \]

### Step 5: Set up the equation
Substituting \(\tan(60^\circ)\) into the equation gives us:
\[ \sqrt{3} = \frac{H}{4.2} \]

### Step 6: Solve for \(H\)
To find \(H\), we can rearrange the equation:
\[ H = 4.2 \cdot \sqrt{3} \]

### Step 7: Calculate \(H\)
Now, we can calculate \(H\):
\[ H \approx 4.2 \cdot 1.732 \approx 7.2734 \text{ m} \]

### Step 8: Round to the nearest value
The height \(H\) is approximately \(7.27\) m, which we can round to \(7.3\) m.

### Conclusion
Thus, the height of the point where the ladder touches the window from the ground is closest to \(7.3\) m.

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