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?
Pretty straightforward question, but it gives a messy diagram, and so, I can't make anything out of it :L
Also big thanks to Damon for helping me out with all these bearing questions :)
first leg
north +1.5 cos 30
east +1.5 sin 30
second leg
north -3.5 cos 20
east +3.5 sin 20
third leg
north +6.25 sin 30
east -6.25 cos 30
so where are they from start?
north 1.5 cos 30 - 3.5 cos 20 + 6.25 sin 30
east +1.5 sin 30 +3.5 sin 20 - 6.25 cos 30
d = Total distance = sqrt(north^2+east^2)
A = angle clockwise from north (compass true angle, not usual math angle)
tan A = east/north
pick the right quadrant by the signs of north and east distances
to get back to origin go distance d at A + 180 if A <180 or at A - 180 if A>180
When using Bearings, multiplying the distance by the Cos of the angle gives the vertical component instead of the usual hor. component.
Displacement = 1.5km[30o] + 3.5[160] + 6.25[300].
Disp. = (1.3i+0.75) + (-3.29i+1.2) + (3.125i-5.41) = -3.46 + 1.14i = 3.64km[-18.2o] = 3.64km[288.2o].
Direction: 3.64km at a bearing of 288o.
Well, it sounds like these bushwalkers have taken quite the adventurous route! Let's see if we can navigate our way back to their base.
To find the total displacement, we need to add up the distances and directions they traveled.
First, they traveled 1.5 km at a true bearing of 030°. Think of this as moving 1.5 km in the direction of a clock at 3 o'clock.
Next, they traveled 3.5 km at a true bearing of 160°. Imagine facing a clock with 12 o'clock straight ahead, and then moving towards the 4 o'clock position.
Finally, they traveled 6.25 km at a true bearing of 300°. Picture facing the same clock, and this time moving towards the 10 o'clock position.
Now it's time to bring out our trusty compass! Adding up the bearings, we have 030° + 160° + 300° = 490°.
But wait! We need to make sure our bearing is less than 360°, so let's reduce it. 490° - 360° = 130°.
So the bushwalkers need to travel a total distance of 1.5 km + 3.5 km + 6.25 km = 11.25 km at a true bearing of 130° to return to their base.
Now, I hope the base has some cold drinks and comfy chairs waiting for them because it sounds like they've been on quite the journey! Safe travels to those adventurous bushwalkers!
To find the return distance and true bearing, we can break down the movements of the bushwalkers into horizontal (east-west) and vertical (north-south) components.
First, let's determine the horizontal and vertical components of the individual movements:
Movement 1:
Distance: 1.5 km
True bearing: 030°
Horizontal component of Movement 1:
cos(30°) * 1.5 km = 1.299 km (east)
Vertical component of Movement 1:
sin(30°) * 1.5 km = 0.75 km (north)
Movement 2:
Distance: 3.5 km
True bearing: 160°
Horizontal component of Movement 2:
cos(160°) * 3.5 km = -3.409 km (west)
Vertical component of Movement 2:
sin(160°) * 3.5 km = 2.948 km (north)
Movement 3:
Distance: 6.25 km
True bearing: 300°
Horizontal component of Movement 3:
cos(300°) * 6.25 km = 5.406 km (west)
Vertical component of Movement 3:
sin(300°) * 6.25 km = -3.125 km (south)
Now, let's calculate the total horizontal and vertical displacements:
Total horizontal displacement:
1.299 km - 3.409 km + 5.406 km = 3.296 km (east)
Total vertical displacement:
0.75 km + 2.948 km - 3.125 km = 0.573 km (north)
The horizontal and vertical displacements give us a right-angled triangle. To find the distance and bearing of the return path to the base, we can use the Pythagorean theorem and trigonometry.
The distance can be found using the Pythagorean theorem:
Distance = √(horizontal displacement^2 + vertical displacement^2)
Distance = √(3.296 km^2 + 0.573 km^2)
Distance = √(10.857 km^2)
Distance = 3.295 km (rounded to 3 decimal places)
The true bearing can be found using trigonometry:
True bearing = arctan(vertical displacement / horizontal displacement)
True bearing = arctan(0.573 km / 3.296 km)
True bearing = arctan(0.174)
True bearing = 10.239° (rounded to 3 decimal places)
Therefore, the bushwalkers need to travel approximately 3.295 km at a true bearing of 010° to return to their base.
Correction: Tan A = X/Y = (-3.46)/1.14 = -3.03509.
A = -72o = 72o N. of W. = 342o Clockwise from +y-axis
Direction: 3.64km at a bearing of 342o.