balloon is directly above one end of bridge. The angle of depression of the other end of the bridge from the balloon is `48"^(@).` If the height of the balloon above the bridge is 122 m, then what is the length of the bridge?

Options

A. `122 sin 48^(@) m `
B. `122 tan 42 ^(@) m `
C. `122 cos 48 ^(@) m`
D. `122 tan 48^(@) m`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have a balloon directly above one end of a bridge. The height of the balloon above the bridge is given as 122 meters. The angle of depression from the balloon to the other end of the bridge is 48 degrees. We need to find the length of the bridge.

### Step 2: Draw a Diagram
Draw a right triangle where:
- Point A is the position of the balloon.
- Point B is the point directly below the balloon on the bridge.
- Point C is the other end of the bridge.
- The height AB = 122 m (height of the balloon above the bridge).
- The angle of depression from A to C is 48 degrees.

### Step 3: Identify the Angles
Since the angle of depression from A to C is 48 degrees, the angle of elevation from C to A is also 48 degrees (alternate interior angles).

### Step 4: Use Trigonometric Ratios
In triangle ABC:
- AB is the opposite side to angle A (which is 48 degrees).
- BC is the adjacent side to angle A (which we need to find).

Using the cotangent function:
\[ \cot(48^\circ) = \frac{\text{Adjacent (BC)}}{\text{Opposite (AB)}} \]
Substituting the known values:
\[ \cot(48^\circ) = \frac{BC}{122} \]

### Step 5: Rearranging the Equation
Rearranging the equation to find BC:
\[ BC = 122 \cdot \cot(48^\circ) \]

### Step 6: Calculate the Value
Now, we need to calculate \( \cot(48^\circ) \).
Using a calculator:
\[ \cot(48^\circ) \approx 1.1106 \]
Now substituting this value back into the equation:
\[ BC = 122 \cdot 1.1106 \approx 135.5 \text{ meters} \]

### Final Answer
The length of the bridge (BC) is approximately **135.5 meters**.

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