A 22 m long ladder (whose foot is on the ground) leans against a wall making an angle of `60^(@)` with the wall. What is the height (in m) of the point where the ladder touches the wall from the ground?

Options

A. `(22sqrt(2))/(3)`
B. `11sqrt(2)`
C. 11
D. `11sqrt(3)`

Answer

To solve the problem step by step, we can use trigonometric functions. Here’s how to find the height at which the ladder touches the wall:

### Step 1: Understand the problem
We have a ladder leaning against a wall, forming a right triangle with the ground and the wall. The length of the ladder is the hypotenuse, and we need to find the height where the ladder touches the wall (the opposite side of the angle).

### Step 2: Identify the given values
- Length of the ladder (hypotenuse) = 22 m
- Angle with the wall (θ) = 60°

### Step 3: Use the sine function
In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, we can write:
\[ \sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Here, the opposite side is the height (h) where the ladder touches the wall, and the hypotenuse is the length of the ladder.

### Step 4: Substitute the known values into the sine function
\[ \sin(60°) = \frac{h}{22} \]
We know that \(\sin(60°) = \frac{\sqrt{3}}{2}\).

### Step 5: Set up the equation
Substituting the value of \(\sin(60°)\) into the equation gives:
\[ \frac{\sqrt{3}}{2} = \frac{h}{22} \]

### Step 6: Solve for h
To find h, we can rearrange the equation:
\[ h = 22 \cdot \frac{\sqrt{3}}{2} \]
Calculating this:
\[ h = 11\sqrt{3} \]

### Step 7: Calculate the numerical value of h
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ h \approx 11 \cdot 1.732 \approx 19.052 \]
Thus, the height at which the ladder touches the wall is approximately 19.052 m.

### Final Answer
The height of the point where the ladder touches the wall from the ground is \(11\sqrt{3} \approx 19.052\) m.

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