A ladder 15 metres long just reaches the top of a vertical wall. If the ladder makes an angle of `60^(@)` with the wall, find the height of the wall.

Options

A. `6.5`m
B. `7.5`m
C. `8.5`m
D. `9.5`m

Explanation

To find the height of the wall using the given information, we can use trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a ladder that is 15 meters long leaning against a vertical wall, forming a right triangle with the wall and the ground. The angle between the ladder and the wall is 60 degrees.

### Step 2: Identify the Triangle
In the right triangle formed:
- The ladder (15 meters) is the hypotenuse (AB).
- The height of the wall (h) is the opposite side (BC).
- The distance from the base of the wall to the foot of the ladder is the adjacent side (AC).

### Step 3: Use the Sine Function
We can use the sine function to find the height of the wall. The sine of an angle in a right triangle is given by the ratio:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
Here, \(\theta = 60^\circ\), the opposite side is the height of the wall (h), and the hypotenuse is the length of the ladder (15 m).

### Step 4: Set Up the Equation
Using the sine function:
\[ \sin(60^\circ) = \frac{h}{15} \]

### Step 5: Calculate \(\sin(60^\circ)\)
We know that:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Now substitute this value into the equation:
\[ \frac{\sqrt{3}}{2} = \frac{h}{15} \]

### Step 6: Solve for h
To find the height (h), multiply both sides by 15:
\[ h = 15 \cdot \frac{\sqrt{3}}{2} \]
\[ h = \frac{15\sqrt{3}}{2} \]

### Step 7: Calculate the Numerical Value
Now, we can calculate the numerical value of \(h\):
\[ h \approx \frac{15 \cdot 1.732}{2} \approx \frac{25.98}{2} \approx 12.99 \text{ meters} \]

### Conclusion
The height of the wall is approximately \(12.99\) meters.

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