Question 1120134: Hello,
I am facing problem with this question-
During an expedition, the leader of a scout troop decided to save time by hiking around a wooded area between the base camp and a lake. The troop reached the lake by walking 30 minutes on a bearing of N60degree East, then 45 minutes on a bearing of N80degree West. (Our teacher gave this as 280 degree, and I think this is a typo error). The leader figured that they walked at an average speed of 3km/h. If they could have walked straight through the forested area at 1km/h, did the troop save any time by going around? If so, how much?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! well, there's a big difference between north 80 degrees west and north 280 degrees west.
north 280 degrees west would be the same as north 80 degrees east which doesn't make much sense, unless your instructor is deliberately trying to trick you.
assuming you are correct, then, once all is settled, you will wind up with triangle ABC as shown in the diagram below.
side a is opposite angle A.
side b is opposite angle B.
side c is opposite angle C.
angle B is equal to 40 degrees.
length of side a = 45/60 * 3 = 2.25 kilometers.
length of side c = 30/60 * 3 = 1.5 kilometers.
length of side b is what you want to find.
since the speed is in kilometers per hour, we will translate the time to hours.
45 minutes is equal to 45/60 hours.
30 minutes is equal to 30/60 hours.
you can find the length of side b by invoking the law of cosines.
the triangle has been labeled the way it is to make translating to the law of cosines easier.
the formula to use is b = sqrt(a^2 + c^2 - 2 * a * c * cos(B))
this formula becomes b = sqrt(2.25^2 + 1.5^2 - 2 * 2.25 * 1.5 * cos(40)).
solve for b to get b = 1.46345481 kilometers.
since this distance is traveled at 1 kilometer per hour, and since 1 kilometer per hour is equal to 1/60 kilometers per minute, divide that by 1/60 to get the number of minutes.
you get 1.46345481 / (1/60) = 87.80728917.
if the troop went through the woods, it would have taken them 87.80728917 minutes.
it took them 75 minutes to go around the woods.
they saved 87.80728917 minus 75 = 12.80728917 minutes by going around.
the top diagram is the final triangle formed.
this is the final triangle that allowed us to use the law of cosines to find the length of side b.
the bottom diagram shows how angle B was determined to be equal to 40 degrees.
AB is 60 degrees north east.
triangle ABD is formed by drawing vertical line from B to D.
that makes triangle ABD a right triangle.
angle A of this triangle is 30 degrees because it is complement to 60 degrees.
angle D of this triangle is 90 degrees because it is the intersection of a vertical and horizontal line.
angle B of this triangle is 60 degrees because sum of the angles of a triangle is 80 degrees.
angle B of triangle ABC is 40 degrees because 80 + 40 + 60 = 180 degrees.
the other angles of triangle ABC can also be derived through either the use of the law of cosines or the law of sines.
i used an online law of cosines calculator at https://www.calculatorsoup.com/calculators/geometry-plane/triangle-law-of-cosines.php
it found the other two angles for me, saving me a lot of additional work that i didn't feel like doing.
the results are shown below:
in solving these types of problems, you will more then likely make use of the law of cosines or the law of sines.
in this problem, the law of cosines was used after find that angle B was 40 degrees
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