The instructions give these facts
- Fact 1: They walk for 30 minutes (0.5 hrs) on a bearing of N60E
- Fact 2: They walk for 45 minutes (0.75 hrs) on a bearing of N80W
- Fact 3: The average walking speed is 3 km/h
Let's set up the diagram.
Step 1) Plot point A as the starting point. Draw a faint dashed line segment straight north of this point to set up a guideline. Then draw a 60 degree angle off this vertical segment to represent the bearing of N60E. This is the left most red angle in the diagram below. This is angle DAB.
For more information about compass bearings, see this page below.
http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html
Step 2) Following this bearing, draw a line segment and mark it to be 1.5 km. This value comes from fact 1 and fact 3. Distance = rate*time = 3*0.5 = 1.5; Mark point B at the other end of the segment.
Step 3) Draw a vertical dashed line through point B. Make sure to extend the line above and below point B. Plot points E and F at the endpoints of this dashed segment. I have point E north of B, while F is south of B. Angle ABF is 60 degrees because EF is parallel to AD, also because angles DAB and ABF are alternate interior angles.
Step 4) Mark angle EBC to be 80 degrees. With point B as the reference point, look directly north to point E, then turn 80 degrees to the west to look at point C. This is the bearing N80W.
Step 5) Walk 2.25 km from B to C. The 2.25 comes from fact 2 and fact 3. Distance = rate*time = 3*0.75 = 2.25
Step 6) The red and blue angles will help us find the green angle (see diagram below). The three angles add to 180. If x is the green angle, then 80+x+60 = 180 solves to x = 40. Therefore, angle ABC is 40 degrees, which is the green angle.
After those steps, you should have this triangle set up
Let's clean up a bit of the extra unneeded segments and points.
We also don't need the compass anymore.
Sides:
a = 2.25 km
b = unknown
c = 1.5 km
Angles
A = unknown
B = 40 degrees
C = unknown
Use the Law of Cosines to solve for side 'b'
b^2 = a^2 + c^2 - 2*a*c*cos(B)
b^2 = 2.25^2 + 1.5^2 - 2*2.25*1.5*cos(40)
b^2 = 5.0625 + 2.25 - 6.75*cos(40)
b^2 = 5.0625 + 2.25 - 6.75*0.766044443118978
b^2 = 5.0625 + 2.25 - 5.1707999910531
b^2 = 2.1417000089469
b = sqrt(2.1417000089469)
b = 1.46345481957828
b = 1.46345
Side b is roughly 1.46345 km
So the distance from A to C is roughly 1.46345 km.
Walking at a speed of 1 km/hr through the forest, it will take roughly t = d/r = 1.46345/1 = 1.46345 hours to go from A to C.
If we went from A to B then to C, then it would take
(30 min)+(45 min) = (0.5 hr)+(0.75 hr) = 1.25 hr
The time difference is:
1.46345 - 1.25 = 0.21345 hours
0.21345 hours = (0.21345 hours)*(60 min/1 hr)
0.21345 hours = 12.807 minutes
12.807 minutes = 12 min + 0.807 min
12.807 minutes = 12 min + (0.807 min)*(60 sec/1 min)
12.807 minutes = 12 min + 48.42 seconds
12.807 minutes = 12 min, 48.42 seconds
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Answer: Yes the troop saved time by going around the forest.
They saved about 12.807 minutes (equivalent to 12 min, 48.42 seconds).