Understanding the Balloon and Bridge Geometry

The problem describes a scenario involving a balloon, a bridge, and an angle of depression. We are given the height of the balloon above the bridge and the angle of depression from the balloon to the far end of the bridge. We need to find the length of the bridge.

Let's visualize this scenario as a right-angled triangle. Assume:

• Point B is the position of the balloon.
• Point A is the end of the bridge directly below the balloon.
• Point C is the other end of the bridge.

The line segment BA represents the height of the balloon above the bridge, which is given as 122 m. The line segment AC represents the length of the bridge, which we need to find. Since A is directly below the balloon and the bridge is horizontal, the angle BAC is 90 degrees. Thus, triangle BAC is a right-angled triangle at A.

Angle of Depression and Its Relation to Triangle Angles

The angle of depression from the balloon (B) to the other end of the bridge (C) is given as 48°. The angle of depression is the angle between the horizontal line from the observer (balloon) and the line of sight to the object (point C on the bridge). Let's draw a horizontal line BX through B. The angle of depression is $\angle XBC = 48^\circ$.

Since the horizontal line BX is parallel to the bridge (line AC), the line segment BC is a transversal intersecting parallel lines. Therefore, the angle of depression ($\angle XBC$) is equal to the angle of elevation from C to B ($\angle BCA$) because they are alternate interior angles. So, $\angle BCA = 48^\circ$.

Using Trigonometry to Find the Bridge Length

Now we have a right-angled triangle ABC, where:

• BA (Height) = 122 m
• $\angle BCA$ (Angle at C) = 48°
• AC (Length of bridge) is what we need to find.

In triangle ABC, with respect to the angle at C (48°), BA is the side opposite to the angle, and AC is the side adjacent to the angle. The trigonometric ratio that relates the opposite side and the adjacent side is the tangent function.

So, we can write:

$\tan(\angle BCA) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{BA}{AC}$

Substituting the known values:

$\tan(48^\circ) = \frac{122}{AC}$

To find AC, we can rearrange the equation:

$AC = \frac{122}{\tan(48^\circ)}$

While this is a correct expression for the length of the bridge, let's look at the options. The options are in the form of $122 \times (\text{trigonometric function of an angle})$. Let's consider the other acute angle in the right-angled triangle ABC, which is the angle at B ($\angle ABC$). The sum of acute angles in a right triangle is 90°.

$\angle ABC + \angle BCA = 90^\circ$

$\angle ABC + 48^\circ = 90^\circ$

$\angle ABC = 90^\circ - 48^\circ = 42^\circ$

Now, let's use the angle at B ($\angle ABC = 42^\circ$). With respect to this angle, AC is the side opposite, and BA is the side adjacent. Using the tangent function again:

$\tan(\angle ABC) = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{AC}{BA}$

Substituting the values:

$\tan(42^\circ) = \frac{AC}{122}$

Now, solve for AC:

$AC = 122 \times \tan(42^\circ)$

This expression matches one of the given options.

Summarizing the Trigonometric Relationships

From the second relationship, we find the length of the bridge AC = $122 \tan(42^\circ)$ m.

Final Answer Calculation

Based on our trigonometric analysis using the angle $42^\circ$, the length of the bridge is $122 \tan(42^\circ)$ meters.

Revision Table: Bridge Length Calculation

Additional Information: Trigonometry and Angles

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. It is particularly useful for solving problems involving heights, distances, and angles that form right-angled triangles.

• Angle of Elevation: The angle measured upwards from a horizontal line to a point above the observer.
• Angle of Depression: The angle measured downwards from a horizontal line to a point below the observer.
• Alternate Interior Angles: When two parallel lines are intersected by a transversal, the angles on opposite sides of the transversal and between the parallel lines are equal. This is a key concept in relating the angle of depression/elevation outside a triangle to an angle inside the triangle.
• Tangent Function: For an acute angle $\theta$ in a right-angled triangle, the tangent of $\theta$ is defined as the ratio of the length of the side opposite $\theta$ to the length of the side adjacent to $\theta$. $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$.

Understanding these concepts helps in setting up the correct trigonometric equation to solve height and distance problems.