WPP #13 and 14: Unit P Concepts 6 & 7: Applications with Law of Sines & Law of Cosines

On a cool December morning, David decided to go skiing out on the alps. He knew that his buddy Thomas had skii lodge and equipment. So Thomas welcomed David and the two went skiing. After a few hours  they decided to head back, but they had gone so far off course, they were lost. Thomas called the Police station, but the heavy blizzard was interfering with the connection. The police needed the two friends exact location but it was so cold and snowy they couldn't give an exact location. Police station B is due south of Station A and they are 100 miles apart. From what David can tell, they are S 45 degrees W of Station A, and N 65 degrees W of Station B. How far is each station to the lonely skiiers?

Here is how the triangle should look like. This a ASA problem since we give two angles and one side in between them.  Since we give two angles ( 65 and 45), we add them up which equals 110 degrees. Then we subtract that value from 180 degrees, to find our missing angle value, of 70 degrees.

By using the Law of Sine to solve for side "a". Angle C, which is where the skiiers are, will be our bridge to solve for sides "a' and side "b". Sin 70/ 100 = Sin 45/ a, we can cross multiply to give us a(sin 70) = 100(sin 450). Then we divide sin 70 to both sides to cancel sin 70 on one side so that it will be a = 100(sin 45)/ sin 70. Station A is 75.2 miles away.

As said before, sin 70/ 100 is our next bridge and it will be used to fine side "b". So sin 70/100 = sin 65/b, we cross multiply to give us b(sin 70) = 100(sin 65). We then divide sin 70 to both sides so that sin 70 cancels on one side so that we can have b = 100(sin 65)/ sin 70. Station B is 96.4 miles away from the skiiers.
Once they got rescued and taken back to Police station A they are taken back home. David and Thomas both leave the station at the same time, they diverge an angle of 100 degrees. If David is 4 miles away from the station and Thomas is 5.5 miles away, then how far away are they from each other?

This picture hopefully sums up the rest of the work. We will use the Law of Cosine to find the missing side. Make sure all the work is right up to this point! Then you write your equation as a^2 = 4^2 + 5.5^2 - 2(4)(5.5) cos 100. After that, we just plug that in to our calculator and we get our answer of 7.3 miles. David and Thomas are both 7.3 miles apart.

CONGRATULATIONS!!:D You have solved the problem {=~^0^~=}