From a balloon vertically above a straight road, the angle of depression of two cars at an instant are found to be `45^(@)` and `60^(@)`. If the cars are 100 m apart, find the height of the balloon.

Answer

To solve the problem step by step, we will use the concepts of trigonometry, particularly the tangent function, as we are dealing with angles of depression.

### Step 1: Understand the Geometry
We have a balloon at point A, which is vertically above a straight road. The two cars are at points C and D, with the distance between them (CD) being 100 m. The angles of depression from the balloon to the cars are 60° for car C and 45° for car D.

### Step 2: Draw the Diagram
1. Draw a vertical line representing the height of the balloon (AB).
2. Draw a horizontal line representing the road (CD).
3. Mark points C and D on the road, with CD = 100 m.
4. Draw lines from A to C and A to D, making angles of depression of 60° and 45°, respectively.

### Step 3: Set Up the Triangles
- In triangle ABC (where angle ACB = 60°):
- The height of the balloon (AB) is the opposite side.
- The distance from point B to point C (BC) is the adjacent side.

Using the tangent function:
\[ \tan(60°) = \frac{AB}{BC} \]
\[ \sqrt{3} = \frac{H}{BC} \quad \text{(where H is the height of the balloon)} \]
Thus, we can express BC in terms of H:
\[ BC = \frac{H}{\sqrt{3}} \quad \text{(Equation 1)} \]

- In triangle ABD (where angle ADB = 45°):
- The height of the balloon (AB) is again the opposite side.
- The distance from point B to point D (BD) is the adjacent side, which is equal to BC + CD (where CD = 100 m).

Using the tangent function:
\[ \tan(45°) = \frac{AB}{BD} \]
\[ 1 = \frac{H}{BD} \]
Thus, we can express BD in terms of H:
\[ BD = H \quad \text{(Equation 2)} \]

### Step 4: Relate BD to BC and CD
From the previous step, we know:
\[ BD = BC + CD \]
Substituting Equation 1 into this:
\[ H = \frac{H}{\sqrt{3}} + 100 \]

### Step 5: Solve for H
Rearranging the equation:
\[ H - \frac{H}{\sqrt{3}} = 100 \]
Taking H common:
\[ H \left(1 - \frac{1}{\sqrt{3}}\right) = 100 \]
Calculating \(1 - \frac{1}{\sqrt{3}}\):
\[ 1 - \frac{1}{\sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}} \]
Thus, we have:
\[ H \cdot \frac{\sqrt{3} - 1}{\sqrt{3}} = 100 \]
Now, solving for H:
\[ H = \frac{100 \cdot \sqrt{3}}{\sqrt{3} - 1} \]

### Step 6: Calculate the Height
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[ H \approx \frac{100 \cdot 1.732}{1.732 - 1} = \frac{173.2}{0.732} \approx 236.5 \text{ m} \]

### Final Answer
The height of the balloon is approximately **236.5 m**.