A Coast Guard cutter detects an unidentified ship at a distance of 19.0 km in the direction 15.0° east of north. The ship is traveling at 24.0 km/h on a course at 40.0° east of north. The Coast Guard wishes to send a speedboat to intercept and investigate the vessel.

(a) If the speedboat travels at 48.0 km/h, in what direction should it head? Express the direction as a compass bearing with respect to due north.

To solve this problem, we need to determine the direction in which the speedboat should head in order to intercept the ship. This involves analyzing the relative motion between the Coast Guard cutter (the origin point), the unidentified ship, and the speedboat. Let's proceed step by step.

### Step 1: Break down the motion into components
First, let's break down the velocity components of the unidentified ship and the speedboat in terms of east and north directions.

#### Ship's velocity components:
- **Ship's speed**: 24.0 km/h
- **Direction of ship's movement**: 40.0° east of north

We can break this into north and east components using trigonometry:

- North component: \( v_{\text{north, ship}} = 24.0 \cdot \cos(40.0^\circ) \)
- East component: \( v_{\text{east, ship}} = 24.0 \cdot \sin(40.0^\circ) \)

#### Speedboat's velocity components:
- **Speedboat's speed**: 48.0 km/h
- The speedboat's direction is what we want to find.

Let's assume the speedboat travels at an angle \( \theta \) east of north. We will break its velocity into components as well:

- North component: \( v_{\text{north, speedboat}} = 48.0 \cdot \cos(\theta) \)
- East component: \( v_{\text{east, speedboat}} = 48.0 \cdot \sin(\theta) \)

### Step 2: Relative motion
Since the speedboat is trying to intercept the ship, its relative velocity must point toward the initial position of the ship. The relative velocity in both the north and east directions must balance the position of the ship.

#### Ship's initial position:
- The ship is located 19.0 km away from the cutter.
- The direction is 15.0° east of north.

We can find the components of the initial position of the ship in the north and east directions:

- North component: \( d_{\text{north, ship}} = 19.0 \cdot \cos(15.0^\circ) \)
- East component: \( d_{\text{east, ship}} = 19.0 \cdot \sin(15.0^\circ) \)

### Step 3: Set up the system of equations
We now have the position and velocities of both the ship and the speedboat. To find the direction \( \theta \) for the speedboat, we can set up the following system of equations based on the requirement that the relative position of the ship and the speedboat changes such that the speedboat intercepts the ship:

1. In the north direction:
   \[
   48.0 \cdot \cos(\theta) - 24.0 \cdot \cos(40.0^\circ) = \frac{19.0 \cdot \cos(15.0^\circ)}{t}
   \]
   
2. In the east direction:
   \[
   48.0 \cdot \sin(\theta) - 24.0 \cdot \sin(40.0^\circ) = \frac{19.0 \cdot \sin(15.0^\circ)}{t}
   \]

### Step 4: Solve for \( \theta \)
Let me calculate the necessary values and find the angle \( \theta \).The speedboat should head at an angle of approximately **27.5° east of north** in order to intercept the ship.

Would you like more details on the solution, or do you have any further questions?

Here are some related questions:
1. How would the solution change if the ship were traveling at a different speed?
2. What if the speedboat's speed was only 30 km/h instead of 48 km/h?
3. How can we calculate the time it will take for the speedboat to intercept the ship?
4. How would the result change if the ship’s course was directly north instead of 40° east of north?
5. Can you calculate the distance the speedboat will travel during the interception?

**Tip:** When analyzing relative motion, breaking velocities into components helps manage direction and magnitude simultaneously.

A Coast Guard cutter detects an unidentified ship at a distance of 19.0 km in the direction 15.0° east of north. The ship is traveling at 24.0 km/h on a course at 40.0° east of north. The Coast Guard wishes to send a speedboat to intercept and investigate the vessel. If the speedboat travels at 48.0 km/h, in what direction should it head? Express the direction as a compass bearing with respect to due north.

Explore how to solve a relative motion problem involving a Coast Guard speedboat intercepting a ship. This problem uses trigonometry and vector decomposition to determine the direction in which the speedboat should head. Suitable for Grades 11-12, the solution includes key concepts like relative velocity and the law of cosines.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation