Question

A rotating light on top of a lighthouse sends out rays of light in opposite directions. As the beacon rotates, the ray at angle \(\theta\) makes a spot of light that moves along the shore. The lighthouse is located \(500 \mathrm{m}\) from the shoreline and makes one complete rotation every 2 min. a) Determine the equation that expresses the distance, \(d,\) in metres, as a function of time, \(t,\) in minutes. b) Graph the function in part a). c) Explain the significance of the asymptote in the graph at \(\theta=90^{\circ}\).

Short answer

The equation is \(d(t) = 500 \tan(180t)\). The function has vertical asymptotes where \(t = \frac{1}{2} + k\).

Step-by-step solution

Understand the problem

The lighthouse is 500 meters from the shoreline, and the rotating beacon completes one rotation every 2 minutes. We need to find the distance, \(d\), as a function of time, \(t\).

Relate angle and time

Since the lighthouse completes one rotation in 2 minutes and one rotation corresponds to \(360^{\text{°}}\), the angle \(\theta\) at any time \(t\) can be expressed as \(\theta = 180t\) degrees (since \(\theta\) completes half a rotation in 1 minute).

Use trigonometric relationships

The distance \(d\) from the lighthouse to a point on the shore based on the angle \(\theta\) can be found using the tangent function: \[ d = 500 \tan(\theta). \]

Substitute for \(\theta\)

Substitute \(\theta = 180t\) into the tangent function to express the distance \(d\) as a function of time \(t\): \[ d(t) = 500 \tan(180t). \]

Graph the function

To graph the function \(d(t) = 500 \tan(180t)\), plot \(t\) on the x-axis and \(d\) on the y-axis. Keep in mind that the tangent function has vertical asymptotes where \(\theta = 90^{\text{°}} + k360^{\text{°}}\).

Explain the asymptote

The significance of the asymptote at \(\theta = 90^{\text{°}}\) and every \(180^{\text{°}}\) thereafter (i.e., at \(t = \frac{1}{2} + k\)) is that the distance \(d\) approaches infinity because \(\tan(\theta)\) approaches infinity at these angles, indicating the light ray is parallel to the shoreline.

Key concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In this lighthouse problem, we use these functions to relate the angle the light makes with time and distance. The main trigonometric functions include sine, cosine, and tangent. Here, we focus on the tangent function because it helps us find the distance along the shore.

Each trigonometric function has its own unique properties. For example:

• Sine relates the opposite side to the hypotenuse.
• Cosine relates the adjacent side to the hypotenuse.
• Tangent relates the opposite side to the adjacent side.

Understanding these relationships is crucial for solving problems involving triangles, rotations, and waves, as seen in the lighthouse problem.

Tangent Function

The tangent function, often written as \(\tan(θ)\), is one of the primary trigonometric functions. It describes the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. For this lighthouse problem, we encounter the tangent function as follows:

\[ \tan(θ) = \frac{\text{opposite}}{\text{adjacent}} \]
In our case, the 'opposite' side is the distance from the lighthouse to the point on the shore, and the 'adjacent' side is the fixed distance from the lighthouse to the shoreline, which is 500 meters.

This relationship becomes: \[ d = 500 \tan(\theta) \]
By substituting \( \theta = 180t \), where \( θ \) is the angle in degrees changing over time \(t \) in minutes, we get the distance function:

\[ d(t) = 500 \tan(180t) \]
This function gives us the distance \(d\) along the shoreline at any given time \( t \).

Periodic Functions

Periodic functions are functions that repeat their values at regular intervals. In this exercise, the function \( \tan(180t) \) is periodic.

Key features of periodic functions include:

• The period is the length of one complete cycle of the function.
• Examples include sine, cosine, and tangent functions, which repeat their values over specific intervals.

For the lighthouse problem:

Since the lighthouse completes one full rotation every 2 minutes, our function \( \tan(180t) \) will repeat its pattern every 2 minutes. This periodicity is significant because it helps predict the light's location along the shore at any given moment. Understanding the periodic nature of the tangent function allows us to correctly plot and anticipate the distance \( d \) over time.

Vertical Asymptotes

Vertical asymptotes are lines where a function's value approaches infinity. These occur in the tangent function where the angle \( θ \) is 90° + k360° for any integer \( k \).

In the context of this problem, when \( \theta \) equals these critical angles, the tangent function value becomes undefined, causing the distance \( d \) to approach infinity.

Therefore, the function \( d(t) = 500 \tan(180t) \) will have vertical asymptotes where:

\[ 180t = 90° + k360° \]
Solving for \(t \), we get:
\br> \[ t = \frac{1}{2} + k \]
These vertical asymptotes indicate moments when the light beam is parallel to the shoreline, causing the distance to shoot up to infinity. By identifying these points, we can better understand and graph the behavior of our distance function.

Remember, vertical asymptotes are crucial for understanding the limitations and unique behavior of functions like the tangent function.