A ladder is placed along a wall such that its upper end is resting against a vertical wall. The foot of the ladder is 2.4 m from the wall and the ladder is making an angle of `68^(@)` with the ground . Find the height, upto which the ladder reaches.

Answer

To solve the problem step by step, we will use trigonometric ratios. Here’s the solution:

### Step 1: Understand the Problem
We have a ladder leaning against a wall, forming a right triangle with the ground and the wall. We know:
- The distance from the foot of the ladder to the wall (base) = 2.4 m
- The angle between the ladder and the ground = 68°

### Step 2: Identify the Triangle
Let:
- A = the point where the ladder touches the wall (height we want to find)
- B = the foot of the ladder (2.4 m from the wall)
- C = the point where the ladder meets the ground

In triangle ABC:
- AB is the height (h) we want to find.
- BC is the distance from the wall to the foot of the ladder (2.4 m).
- AC is the length of the ladder.

### Step 3: Use the Tangent Function
In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Here, we can use the tangent of the angle (68°):

\[ \tan(68°) = \frac{AB}{BC} \]

Substituting the known values:

\[ \tan(68°) = \frac{h}{2.4} \]

### Step 4: Calculate \(\tan(68°)\)
Using a calculator or trigonometric tables, we find:

\[ \tan(68°) \approx 2.475 \]

### Step 5: Substitute and Solve for h
Now we can substitute \(\tan(68°)\) back into the equation:

\[ 2.475 = \frac{h}{2.4} \]

To find h, multiply both sides by 2.4:

\[ h = 2.475 \times 2.4 \]

### Step 6: Perform the Calculation
Now, we calculate:

\[ h = 2.475 \times 2.4 \approx 5.94 \text{ m} \]

### Conclusion
The height up to which the ladder reaches is approximately **5.94 meters**.

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