Math Problem Statement
: A boat leaves a dock and travels due north at a speed of 10 km/hr. A second boat leaves the same dock at the same time, but travels N50oE at a speed of 15 km/hr. How far apart are the two boats after 2 hours? Assume the second boat broke down after those 2 hours. What direction would the first boat need to travel to pick up the people in the broken down boat.
Solution
Let's solve this problem step by step.
Part 1: Finding the distance between the two boats after 2 hours.
-
Boat 1 (Northbound boat):
- Travels at a speed of 10 km/h due north.
- After 2 hours, the distance traveled by Boat 1 is:
-
Boat 2 (Traveling N50°E):
- Travels at a speed of 15 km/h in the direction N50°E.
- After 2 hours, the distance traveled by Boat 2 is:
-
Setting up a coordinate system:
- The path of Boat 1 is along the positive y-axis (due north).
- The path of Boat 2 makes an angle of 50° with the north direction (N50°E).
Using trigonometry, we can break down the displacement of Boat 2 into components:
- Northward (y-component) is .
- Eastward (x-component) is .
Calculating these components:
-
Calculating the distance between the two boats:
- Boat 1 is at (0, 20) in the coordinate plane.
- Boat 2 is at (22.98, 19.284).
The distance between the two boats is the straight-line distance between these two points:
After 2 hours, the boats are approximately 23.01 km apart.
Part 2: Finding the direction the first boat needs to travel to reach the second boat.
To find the direction Boat 1 must travel to reach Boat 2, we need to calculate the angle between the direction of Boat 1's current position (0, 20) and the broken-down boat's position (22.98, 19.284).
-
Relative position:
- The displacement vector from Boat 1 to Boat 2 is:
- Change in : .
- Change in : .
- The displacement vector from Boat 1 to Boat 2 is:
-
Finding the angle:
- The angle that Boat 1 must travel relative to due east is given by:
This means that Boat 1 must travel 1.79° south of east to reach Boat 2.
Final Answer:
- After 2 hours, the boats are 23.01 km apart.
- Boat 1 must travel 1.79° south of east to reach Boat 2.
Would you like further details on any step or explanation?
Here are five related questions to explore:
- How would the calculation change if Boat 1 traveled at a different speed?
- What if Boat 2 had traveled in a different direction, say N30°E?
- How would you find the time it takes for Boat 1 to reach Boat 2?
- What is the concept of relative velocity, and how would it apply here?
- How would you solve this problem if both boats had variable speeds?
Tip: Always break down vectors into components when solving problems involving angles and directions. It simplifies calculations immensely.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Vector Decomposition
Pythagorean Theorem
Formulas
Distance = speed × time
x-component = distance × sin(angle)
y-component = distance × cos(angle)
Distance between two points = √[(x2 - x1)^2 + (y2 - y1)^2]
tan(θ) = Δy / Δx
Theorems
Pythagorean Theorem
Trigonometric Angle Functions
Suitable Grade Level
Grades 9-12
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