Problem 6 - Calculus: Early Transcendentals

Question

A rotating beacon is located 2 miles out in the water. Let \(A\) be the point on the shore that is closest to the beacon. As the beacon rotates at 10 rev/min, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. How fast is the point where the beam hits the shore moving at an instant when the beam is lighting up a point 2 miles along the shore from the point \(A ?\)

Step-by-Step Solution

Answer

The beam hits the shore moving at \(80\pi\text{ miles per minute}\).

Step 1: Identify the Problem

We have a rotating beacon located at point B, 2 miles from the nearest point A on the shore. The problem asks us to find the speed of the light beam's spot on the shore when it's 2 miles away from point A as the beacon rotates.

Step 2: Model the Problem Mathematically

We can model the problem using a right triangle where B is the beacon, A is the shore point closest to B. If C is the current point on the shore, then AC = 2 miles. The beam's location on the shore relates to angle \(\theta\) which is the angle between the line joining the beacon to the shore point A and the line joining the beacon to point C.

Step 3: Use Trigonometry Relationships

The position of point C on the shore can be expressed in terms of \(\theta\): \(x = 2\tan(\theta)\), where x is the distance from A to C along the shore. When \(x = 2\), we find the specific \(\theta\) value.

Step 4: Derive the Related Rates Equation

Differentiate \(x = 2\tan(\theta)\) with respect to time t to relate the rate of change of x with respect to the rate of change of \(\theta\). This yields \(\frac{dx}{dt} = 2\sec^2(\theta) \frac{d\theta}{dt}\).

Step 5: Substitute Known Values

The beacon rotates at \(10\text{ rev/min} = 20\pi\text{ rad/min}\) which is \(\frac{d\theta}{dt} = 20\pi\). We need \(\sec^2(\theta)\) when \(x = 2\). From \(x = 2\tan(\theta)\), when \(x = 2\), \(\tan(\theta) = 1\), thus \(\theta = \frac{\pi}{4}\), and \(\sec(\theta) = \sqrt{2}\).

Step 6: Calculate \(\frac{dx}{dt}\)

Plug in \(\sec(\theta) = \sqrt{2}\) and \(\frac{d\theta}{dt} = 20\pi\) into the differentiated equation: \(\frac{dx}{dt} = 2(\sqrt{2})^2(20\pi) = 80\pi\).

Step 7: Conclusion

The point where the beam hits the shore is moving at a speed of \(80\pi\text{ miles per minute}\).

Key Concepts

Rotating Beacon Problem

The rotating beacon problem is a fascinating application of calculus and physics, exploring how a beam of light from a rotating beacon interacts with its environment. In this scenario, the beacon is positioned 2 miles away from the closest point on the shore. As it rotates, the light sweeps across the shore. The core objective is to determine the speed at which the spot of light moves along the shore when it is exactly 2 miles from that closest point. This involves several steps: - **Identifying the geometry**: We recognize a triangle formed by the beacon, the nearest shore point, and the light's spot on the shore, which changes positions as the beacon turns. - **Defining movement and position**: The light sweeps across the shore, tracing a path outlined by the spot, which moves at a particular speed depending on the rotational speed of the beacon. The rotating beacon problem is a classic way to delve into the world of related rates, where we analyze how variables are interrelated as they change over time.

Right Triangle Trigonometry

In solving the rotating beacon problem, we're essentially dealing with a right triangle, which calls upon fundamental principles of trigonometry. Consider the triangle formed by:- The beacon as one vertex - The point closest to the beacon on the shore as another vertex- The location on the shore where the beam hits as the third vertex.This triangle helps represent the relation between different distances and angles. Using the properties of this right triangle, we can define the relationship between the angle of the beam and the distance along the shore with the tangent function.

• Tangent function: It helps us express the position of the light beam along the shore using the angle \(\theta\), by x = 2\(\tan(\theta)\) where x is the distance along the shore.
• Identification of angles: As the beacon rotates, the angle \(\theta\) changes, altering the beam's spot position.

Leveraging these trigonometric relationships is crucial for finding how the light moves along the shoreline.

Differentiation with Respect to Time

Differentiation is used to understand how quantities change over time, especially in this rotating beacon problem. By differentiating the trigonometric equation that relates the beam's position on the shore with time, we can find how fast the beam's spot moves.- **Equation Setup**: Start with the equation \(x = 2\tan(\theta)\) that describes the spot's position.- **Differentiation**: Differentiate this equation in terms of time \(t\) to develop a related rates equation. This gives us \(\frac{dx}{dt} = 2\sec^2(\theta) \frac{d\theta}{dt}\).Through this process, we find a connection between the rate at which the beacon's angle changes (\(\frac{d\theta}{dt}\)) and the rate at which the spot of light moves along the shore (\(\frac{dx}{dt}\)). Understanding this dynamic relationship is key to solving related rates problems effectively.

Tangent and Secant Functions

The tangent and secant functions are at the heart of solving the rotating beacon problem. These trigonometric functions help connect the angle of the light beam to the position of its spot on the shoreline.

• Tangent Function: - Defines the relationship between the angle \(\theta\) and the horizontal distance x (\(x = 2\tan(\theta)\)) along the shore. - Provides a way to express how far the light is from its nearest point as it sweeps down the shore.
• Secant Function: - Utilized in the differentiated equation, \(\sec^2(\theta)\) appears in the expression \(\frac{dx}{dt} = 2\sec^2(\theta) \frac{d\theta}{dt}\). - Essential for understanding the rate at which the distance x changes with time due to its relationship with \(\theta\).

By using these two functions, we efficiently establish the mathematical relationships needed to calculate how quickly the spot of light moves along the shore, acting as a bridge between the angle's change and the shoreline distance covered.