Question

The distance from a boat to a bridge is 200 meters. A person aboard measures the angle of elevation to the bridge as \(12^{\circ}\). To the nearest tenth, how far above the water is the bridge?

Short answer

The bridge is approximately 42.5 meters above the water.

Step-by-step solution

Understanding the Problem

We need to find the height of the bridge above the water given two pieces of information: the distance from the boat to the bridge is 200 meters and the angle of elevation from the boat to the bridge is \(12^{\circ}\).

Using the Tangent Function

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, the opposite side is the height of the bridge, and the adjacent side is the distance from the boat to the bridge. The tangent function is therefore \( \tan(12^{\circ}) = \frac{h}{200} \), where \(h\) is the height of the bridge.

Solve for Height

To solve for \(h\), multiply both sides of the equation by 200: \( h = 200 \times \tan(12^{\circ}) \).

Calculate the Tangent

First, calculate \( \tan(12^{\circ}) \). Using a calculator, \( \tan(12^{\circ}) \approx 0.2126 \).

Calculate the Height

Now calculate \(h\) by substituting the tangent value: \( h = 200 \times 0.2126 \). This gives \( h \approx 42.52 \) meters.

Round the Result

Finally, round the result to the nearest tenth: \( h \approx 42.5 \) meters.

Key concepts

Angle of Elevation

The angle of elevation is an important concept in trigonometry, especially when dealing with right triangles. It describes the angle between the horizontal line and the line of sight pointing towards an object above the horizontal. This angle helps us determine how high something is, using basic trigonometric functions.

Imagine standing on a boat and looking up at a bridge. The angle your eyes make with the horizontal line when you look at the top of the bridge is the angle of elevation. In this case, the angle of elevation is given as \(12^{\circ}\).

Understanding the angle of elevation helps us solve problems involving heights and distances. It is closely related to the right triangle concept because it usually forms when you draw lines from the observer to the object and along the horizontal plane. The angle of elevation is always measured from the horizontal upward. This idea brings us to our next important topic, the right triangle.

Right Triangle

A right triangle is a type of triangle where one of the angles is exactly \(90^{\circ}\). Right triangles are extremely useful in trigonometry and geometry because they have simple relationships between their sides and angles that can be easily expressed using trigonometric functions.

In the context of our bridge problem, the scenario forms a right triangle. Here's how:

• The horizontal distance from the boat to the bridge serves as one side of the triangle, known as the adjacent side, and it measures 200 meters.
• The line of sight from the person on the boat to the top of the bridge functions as the hypotenuse.
• The height of the bridge from the water is the opposite side, which is what we want to find.

These sides are associated with the angle of elevation. With the right angle, these elements formulate a right triangle, allowing us to use trigonometric functions like the tangent function to solve for the unknown height.

Tangent Function

In trigonometry, the tangent function is one of the basic functions that connects an angle within a right triangle to the ratio of two of its sides. Specifically, in a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side.

The tangent function is expressed as:\[\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\]In our bridge problem, \(\theta\) is the angle of elevation \(12^{\circ}\), the opposite side (which we want to find) is the height of the bridge, and the adjacent side is the 200-meter distance. This relationship helps us set up the equation:\[\tan(12^{\circ}) = \frac{h}{200}\]Here, \(h\) represents the height of the bridge. Solving for \(h\) involves multiplying both sides by 200. To find \(\tan(12^{\circ})\), we use a calculator:\[\tan(12^{\circ}) \approx 0.2126\]Then, we calculate:\[h = 200 \times 0.2126 \approx 42.52\]Rounding, we find the bridge is approximately 42.5 meters above the water. This process highlights how effectively the tangent function can be used to solve real-world height and distance problems using angles in right triangles.