Math Problem Statement

A ship sailed 3 km west, then 2 km south. Give its three-figure bearings from an observer who stayed at its starting point, correct to the nearest degree. b For a person on the ship, what would be the three-figure bearings looking back to the starting point?

Solution

Part (a): Bearings from the Starting Point

  1. Determine the Final Position of the Ship:

    • The ship first sailed 3 km west, then 2 km south.
  2. Calculate the Angle of the Bearing:

    • The bearing is measured clockwise from the north direction.
    • The ship's final position is to the southwest of the starting point.

    tan(θ)=opposite (south)adjacent (west)=2 km3 km\text{tan}(\theta) = \frac{\text{opposite (south)} }{ \text{adjacent (west)} } = \frac{2 \text{ km}}{3 \text{ km}}

    θ=tan1(23)33.69\theta = \text{tan}^{-1}\left(\frac{2}{3}\right) \approx 33.69^\circ

    • This angle is measured from the westward direction toward the south, so the bearing from the north direction is:

    Bearing=180+33.69214\text{Bearing} = 180^\circ + 33.69^\circ \approx 214^\circ

    • To the nearest degree, the bearing from the starting point is 214°.

Part (b): Bearings from the Ship Back to the Starting Point

  1. Determine the Bearing from the Ship:

    • To find the bearing looking back to the starting point, you simply add 180180^\circ to the bearing found in part (a).

    Reverse Bearing=214+180=394\text{Reverse Bearing} = 214^\circ + 180^\circ = 394^\circ

    • Since bearings are given as three-figure numbers and within 00^\circ to 360360^\circ, subtract 360360^\circ from 394394^\circ.

    Reverse Bearing=394360=34\text{Reverse Bearing} = 394^\circ - 360^\circ = 34^\circ

    • So, the bearing looking back to the starting point from the ship is 034°.

Do you want more details or have any questions about these calculations?

Here are 5 related questions to explore further:

  1. What is the significance of three-figure bearings in navigation?
  2. How would the bearing change if the ship traveled east and then north instead?
  3. How does the Pythagorean theorem apply to find the direct distance between the start and end points?
  4. What would be the bearing if the ship sailed 5 km west and 5 km north?
  5. How are bearings used in aviation compared to marine navigation?

Tip: Always remember that bearings are measured clockwise from the north direction.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Bearings
Right Triangle

Formulas

tan(theta) = opposite / adjacent
Reverse Bearing = Original Bearing + 180° (mod 360°)

Theorems

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Suitable Grade Level

Grades 9-12