A ladder is placed against a vertical tower. If the ladder makes an angle of `30^(@)` with the ground and reached upto a height of 15 m of the tower, find the length of the ladder.

Answer

To solve the problem step by step, we need to use the properties of right triangles and trigonometric ratios. Here’s how we can find the length of the ladder:

### Step 1: Understand the Triangle
We have a right triangle formed by the ladder, the height of the tower, and the ground. The ladder acts as the hypotenuse, the height of the tower (15 m) is the opposite side, and the distance from the base of the tower to the foot of the ladder is the adjacent side.

### Step 2: Identify the Given Values
- Height of the tower (opposite side) = 15 m
- Angle with the ground (θ) = 30°

### 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:
\[ \sin(θ) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Substituting the known values:
\[ \sin(30°) = \frac{15}{\text{Length of the ladder (L)}} \]

### Step 4: Substitute the Value of Sine
We know that:
\[ \sin(30°) = \frac{1}{2} \]
So we can rewrite the equation as:
\[ \frac{1}{2} = \frac{15}{L} \]

### Step 5: Solve for L
To find L, we can cross-multiply:
\[ 1 \cdot L = 2 \cdot 15 \]
\[ L = 30 \]

### Conclusion
The length of the ladder is 30 meters.