Problem 4 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

A light house beacon makes one revolution every two minutes and shines a beam on a straight shore that is one kilometer from the light house. How fast is the beam of light moving along the shore when it is pointing toward the point of the shore closest to the light house? How fast is the beam of light moving along the shore when it is pointing toward a point that is one kilometer from the closest point of the shore to the light house?

Step-by-Step Solution

Answer

The beam moves at \(\pi\) km/min at the closest point and \(2\pi\) km/min at 1 km from it.

Step 1: Understand the Problem

The lighthouse beacon rotates and its light beam moves along the shore. We need to find the speed of the light along the shore, first at the closest point to the lighthouse, and then at a point 1 km from the closest point.

Step 2: Set Up the Scenario

Consider the lighthouse at origin in the coordinate plane, with the shore represented by the x-axis. The closest point on the shore from the lighthouse is at (0,0). The distance from lighthouse to shore is 1 km.

Step 3: Use Geometry to Define Relationships

Let \( \theta \) be the angle the beam makes with a line perpendicular to the shore. Given this, if \( L \) is the distance along the shore from the closest point, then \( L = 1 \tan(\theta) \).

Step 4: Transformation of Angular Velocity

The beacon makes one full revolution every 2 minutes, which translates to an angular velocity \( \frac{d\theta}{dt} = \frac{2\pi}{2} \) radians per minute. This simplifies to \( \pi \) radians per minute.

Step 5: Differentiate with Respect to Time

To find the speed along the shore, differentiate \( L = \tan(\theta) \) with respect to time: \( \frac{dL}{dt} = \sec^2(\theta) \cdot \frac{d\theta}{dt} \).

Step 6: Evaluate the Speed at Closest Point

At the closest point, \( \theta = 0 \) radians. Thus, \( \frac{dL}{dt} = \sec^2(0) \cdot \pi = 1 \cdot \pi = \pi \) km/min.

Step 7: Evaluate the Speed at 1 km from Closest Point

At 1 km along the shore, \( L = 1 \tan(\theta) = 1 \), so \( \theta = \frac{\pi}{4} \) radians. Then, \( \sec^2(\frac{\pi}{4}) = 2 \), and \( \frac{dL}{dt} = 2 \cdot \pi = 2\pi \) km/min.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object makes a full revolution around a point. It is usually described in terms of radians per second or per minute. In the context of the Lighthouse Beacon Problem, we are dealing with the angular velocity of the rotating beacon. The beacon completes one rotation every 2 minutes. This is equivalent to an angular velocity of \( \frac{2\pi}{2} = \pi \) radians per minute.
The formula for calculating angular velocity \( \omega \) is:\[ \omega = \frac{\Delta \theta}{\Delta t} \]
where \( \Delta \theta \) is the change in angle, measured in radians, and \( \Delta t \) is the change in time.
By understanding angular velocity, we can determine how quickly the lighthouse's light beam moves along the shore, based on how quickly it rotates.

Differentiation

Differentiation is a key concept in calculus that deals with finding the rate at which a quantity changes. It's essentially the process of finding a derivative. In this problem, we differentiate with respect to time to find how fast the beam of light moves along the shore.
The function relates \( L \), the distance from the closest point on the shore, using the equation \( L = \tan(\theta) \). To find out how fast \( L \) changes over time, we differentiate both sides with respect to \( t \):

• Differentiate \( L = \tan(\theta) \) to get \( \frac{dL}{dt} = \sec^2(\theta) \cdot \frac{d\theta}{dt} \).

This process allows us to understand the dynamics of how angles and distances are changing in the context of the light shining along the shore.

Trigonometric Functions

Trigonometric functions help to relate angles to lengths. In this problem, the function \( \tan(\theta) \) plays a crucial role because it represents the relationship between the angle \( \theta \) and the distance \( L \) along the shore.
Here, the tangent of the angle gives us the ratio of the opposite side to the adjacent side in a right triangle setup, which describes the beam's position relative to the lighthouse and the shore.
In the problem, when \( \theta \) changes, it affects \( L \) as \( L = 1 \cdot \tan(\theta) \). Thus, knowing \( \tan \), as well as it's derivative \( \sec^2(\theta) \), is crucial to solving how fast \( L \) is changing.
Trigonometry is therefore essential for converting between rotational motion and linear movement along the shore.

Coordinate Geometry

Coordinate geometry is used to attach a physical viewpoint to mathematical concepts. In the Lighthouse Beacon Problem, we consider the lighthouse at the origin on a coordinate plane, with the x-axis representing the straight shoreline. This setup helps in visualizing the problem.
The proximity of the lighthouse to the shoreline is symbolized by coordinates and distances that can be solved using equations like \( L = \tan(\theta) \). This equation represents the position of the light in terms of its distance along the x-axis (the shore).
Ultimately, by using coordinate geometry, we transform the real-world rotation of the beacon into mathematical equations that can be analyzed to find the speed at which the light beam moves along the shore.