Question

Finding the Distance of a Ship from Shore A person in a small boat, offshore from a vertical cliff known to be 100 feet in height, takes a sighting of the top of the cliff. If the angle of elevation is found to be \(30^{\circ},\) how far offshore is the boat?

Short answer

The boat is approximately 173.2 feet offshore.

Step-by-step solution

Identify Known Values

The height of the cliff is 100 feet, and the angle of elevation is 30 degrees.

Set Up the Trigonometric Equation

Use the tangent function, because the tangent of an angle in a right triangle is equal to the opposite side over the adjacent side. Here, the opposite side is the height of the cliff (100 feet), and the adjacent side is the distance from shore that we need to find.

Write the Tangent Equation

The tangent of 30 degrees is given by \[ \tan(30^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{100}{x} \] where \(x\) is the distance from the boat to the shore.

Solve for x

\[ \tan(30^{\circ}) = \frac{100}{x} \] The value of \( \tan(30^{\circ}) \) is \( \frac{1}{\sqrt{3}} \), so \[ \frac{1}{\sqrt{3}} = \frac{100}{x} \] Cross-multiply to solve for \(x\): \[ x = 100 \sqrt{3} \]

Calculate the Final Value

Compute the numerical value: \[ x \approx 100 \times 1.732 \approx 173.2 \text{ feet} \]

Key concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This triangle has one right angle, which makes it different from other triangles.
The two sides that form the right angle are known as the 'legs,' and the side opposite the right angle is called the 'hypotenuse.' This triangle is very useful in many trigonometry problems because it allows us to use the Pythagorean theorem and trigonometric functions.
In the exercise, the triangle formed by the cliff, the boat, and the water's surface is a right triangle. The height of the cliff is one of the legs, and the distance from the boat to the shore is the other leg. The hypotenuse would be the line of sight from the boat to the top of the cliff.

Angle of Elevation

The angle of elevation is the angle formed between the horizontal ground and the line of sight up to an object. It's the angle you look upwards at to see the top of something tall, like a cliff or a building.
In our example, the person in the boat looks up at an angle of elevation of 30 degrees to see the top of the 100-foot cliff.
To find this angle, you'd use a protractor or a similar tool. Knowing the angle of elevation is crucial because it allows us to apply trigonometric functions to solve for unknown distances or heights.

Tangent Function

In trigonometry, the tangent function relates the angles of a right triangle to the lengths of the opposite and adjacent sides. The formula for the tangent of an angle \( \theta \) is:
\[ \tan( \theta ) = \frac{\text{opposite side}}{\text{adjacent side}} \]
This is especially useful in problems where we know one side and an angle and need to find the other side.
In our exercise, the tangent of the 30-degree angle of elevation relates the height of the cliff (opposite side) to the distance from the boat to the shore (adjacent side). We use the equation:
\[ \tan(30^{\text{°}}) = \frac{100}{x} \]
to solve for \( x \), the distance from the boat to the shore.

Solving for Distance

To solve for distance in trigonometry problems involving right triangles, we often use trigonometric functions like sine, cosine, or tangent. For problems involving a right triangle with a given height and angle, the tangent function is often the best choice.
From the previous section, we know that: \[ \tan(30^{\text{°}}) = \frac{100}{x} \]
By knowing that \(\tan(30^{\text{°}}) = \frac{1}{\text{√3}} \), we substitute and solve for \( x \):
\[ \frac{1}{\text{√3}} = \frac{100}{x} \]
Cross-multiplying gives:
\[ x \times \frac{1}{\text{√3}} = 100 \]
Simplifying this, we get:
\[ x = 100 \times \text{√3} \]
Finally, calculating the numerical value:
\[ x \text{ ≈ 100} \times 1.732 \text{ ≈ 173.2 feet} \]
This means the boat is approximately 173.2 feet away from the shore.