Starting from their base in the national park, a group of
bushwalkers travel 1.5 km at a true bearing of 030°, then 3.5 km
at a true bearing of 160°, and then 6.25 km at a true bearing of
300°. How far, and at what true bearing, should the group walk
to return to its base?
arghh, that "bearing" stuff is like chalk on a blackboard to true
mathematicians. Let's translate to the standard trig-angle notation
bearing 30 ---> 60°
bearing 160 ---> 290°
bearing 300 ---> 150°
so we could just use vectors: If P is your final destination,
vector OP = 1.5(cos60,sin60) + 3.5(cos290,sin290) + 6.25(cos150,sin150)
= (-3.4656, 1.1351) , which is in quadrant II
magnitude = √(-3.4656^2 + 1.1351^2) = 3.65 km
angle in real math notation = 162°
converting 162° back to "bearing" would be 288 degrees
BUT we want to walk from P to O, so the bearing should be
288-180 or 108° (the -180 because your turned yourself around)
3.65 km on a bearing of 108°
check my arithmetic
Well, if they want to return to their base, they should probably walk back instead of just standing there. But let's do some calculations for fun, shall we?
First, let's break down their journey. They traveled 1.5 km at a true bearing of 030°, then 3.5 km at a true bearing of 160°, and finally 6.25 km at a true bearing of 300°.
Now, to find the direction they need to head to return to their base, we need to determine the reverse bearings. So, let's reverse the bearings: 030° becomes 210°, 160° becomes 340°, and 300° becomes 120°.
Next, let's add up the distances: 1.5 km + 3.5 km + 6.25 km = 11.25 km.
So, the group needs to walk a total of 11.25 km at a true bearing of 210° to return to their base. But hey, don't forget to stop and enjoy the beautiful scenery along the way!
To find the distance and true bearing for the group to return to their base, we need to calculate the total horizontal and vertical displacement.
Let's break down each leg of their journey and calculate the horizontal and vertical components separately:
First leg:
Distance: 1.5 km
True bearing: 030°
Horizontal component: 1.5 km * sin(30°) = 1.5 km * 0.5 = 0.75 km
Vertical component: 1.5 km * cos(30°) = 1.5 km * √3/2 = 1.3 km (approximately)
Second leg:
Distance: 3.5 km
True bearing: 160°
Horizontal component: 3.5 km * cos(160°) = 3.5 km * -0.94 = -3.29 km
Vertical component: 3.5 km * sin(160°) = 3.5 km * -0.34 = -1.19 km
Third leg:
Distance: 6.25 km
True bearing: 300°
Horizontal component: 6.25 km * cos(300°) = 6.25 km * 0.87 = 5.44 km
Vertical component: 6.25 km * sin(300°) = 6.25 km * -0.5 = -3.13 km
Now, let's calculate the total horizontal and vertical displacement:
Total horizontal displacement = 0.75 km - 3.29 km + 5.44 km = 2.9 km (approximately)
Total vertical displacement = 1.3 km - 1.19 km - 3.13 km = -2.02 km (approximately)
To return to their base, the group needs to walk 2.9 km horizontally and -2.02 km vertically.
The distance can be calculated using the Pythagorean theorem:
Distance to return = √(2.9 km^2 + (-2.02 km)^2) = √(8.41 + 4.08) km = √12.49 km = 3.53 km (approximately)
The true bearing to return can be calculated using the inverse tangent formula:
True bearing to return = arctan(2.02 km / 2.9 km)
Using a calculator or trigonometric table, we find the arctan value to be approximately 34.4°.
Therefore, the group needs to walk approximately 3.53 km at a true bearing of 034° to return to its base.
To find out how far and at what true bearing the group should walk to return to its base, we can use trigonometry and vector addition.
First, let's break down the group's journey into individual components:
1. The first leg of the journey is 1.5 km at a true bearing of 030°.
2. The second leg is 3.5 km at a true bearing of 160°.
3. The third leg is 6.25 km at a true bearing of 300°.
To determine the direction and distance of the return trip, we need to find the resultant vector by adding these individual vectors.
Step 1: Convert the bearings to the corresponding angles in a standard Cartesian plane.
- True bearing of 030° is equivalent to an angle of 360° - 30° = 330°.
- True bearing of 160° is equivalent to an angle of 360° - 160° = 200°.
- True bearing of 300° doesn't require conversion.
Step 2: Convert the angles to radians.
- Angle in radians = Angle in degrees × (π / 180°).
- Angle in radians for 330° = 330° × (π / 180°) = 11π / 6.
- Angle in radians for 200° = 200° × (π / 180°) = 10π / 9.
Step 3: Calculate the x and y components for each leg using trigonometry.
- For the first leg:
- x component = distance * cos(angle)
- x₁ = 1.5 km * cos(11π / 6)
- y component = distance * sin(angle)
- y₁ = 1.5 km * sin(11π / 6)
- For the second leg:
- x component = distance * cos(angle)
- x₂ = 3.5 km * cos(10π / 9)
- y component = distance * sin(angle)
- y₂ = 3.5 km * sin(10π / 9)
- For the third leg:
- x component = distance * cos(angle)
- x₃ = 6.25 km * cos(300°)
- y component = distance * sin(angle)
- y₃ = 6.25 km * sin(300°)
Step 4: Calculate the total x and y components by summing up the individual components.
- Total x component = x₁ + x₂ + x₃
- Total y component = y₁ + y₂ + y₃
Step 5: Calculate the resultant distance and direction using the total x and y components.
- Resultant distance = √(Total x component² + Total y component²)
- Resultant direction = arctan(Total y component / Total x component)
Plug in the values to calculate the resultant distance and direction.