A ship leaves port traveling southwest at a rate of 12 mi/hr. At noon, the ship reaches its closest approach to a radar station, which is on the shore 1.5 mi from the port. If the ship maintains its speed and course, what is the rate of change of the tracking angle θ between the radar station and the ship at 1:30 P.M. (see figure)? (Hint: Use the Law of Sines.) <IMAGE>

Verified step by step guidance

1. First, understand the setup: The ship is traveling southwest, forming a right triangle with the radar station and the path of the ship. The radar station is 1.5 miles from the port, which is the closest point of approach.
2. Define the variables: Let x be the distance the ship has traveled southwest from the closest point of approach. Let θ be the angle between the line from the radar station to the ship and the line from the radar station to the closest point of approach.
3. Use the Law of Sines: In the triangle formed, the Law of Sines states that \( \frac{\sin(\theta)}{x} = \frac{\sin(90^\circ)}{1.5} \). Since \( \sin(90^\circ) = 1 \), this simplifies to \( \sin(\theta) = \frac{x}{1.5} \).
4. Differentiate with respect to time: To find the rate of change of θ, differentiate both sides of the equation \( \sin(\theta) = \frac{x}{1.5} \) with respect to time t. Use the chain rule: \( \frac{d}{dt}[\sin(\theta)] = \cos(\theta) \cdot \frac{d\theta}{dt} \) and \( \frac{d}{dt}[\frac{x}{1.5}] = \frac{1}{1.5} \cdot \frac{dx}{dt} \).
5. Solve for \( \frac{d\theta}{dt} \): Set \( \cos(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{1.5} \cdot \frac{dx}{dt} \). Given that the ship travels at 12 mi/hr, \( \frac{dx}{dt} = 12 \). Substitute this value into the equation and solve for \( \frac{d\theta}{dt} \).

Key Concepts

Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of the triangle. This law is particularly useful in solving problems involving non-right triangles, such as determining unknown angles or side lengths when certain measurements are known.

Rate of Change

The rate of change refers to how a quantity changes in relation to another quantity. In calculus, it is often represented as a derivative, indicating how a function's output value changes as its input value changes. In this context, it helps determine how quickly the tracking angle θ is changing as the ship moves, which is essential for understanding the dynamics of the situation.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. These functions are fundamental in analyzing the relationships between angles and distances in various applications, including navigation and physics. Understanding these functions is crucial for applying the Law of Sines and calculating the tracking angle's rate of change in this problem.