WPP #13&14 Unit P Concepts 6&7

This WPP was made in collaboration with a handsome fellow named Mason Nguyen. Please visit the other awesome posts on his blog at Masonnperiod1.blogspot.com.
1. Law of Sines
       Hannah and her friend Miley are going on a trip to Los Angeles. However, they are at different locations, being 100 miles apart. Hannah flies there in a helicopter at a degree of N 29* E. Miley drives there in a BMW at a degree of N 40* W. They finally arrive at Los Angeles and give each other patty cakes. What was the distance it took for Miley to get to Los Angeles?

 In the above picture, we see the set-up for the problem. Hannah and Miley are 100 miles from each other so thus that is our side value. We draw the north-south line to show our N 29* E angle and our N 40* W angle. x stands for the value which we want to find, which is the distance from Miley's starting point to L.A.

 We find our angles by subtracting the corresponding values we gotten from 90*. We get 61* and 50*. Since we know two values of our angles, we can easily solve for the third by subtracting the two angles from 180*. Therefore, 180-61-50=69. 69* is our third angle. Now that we have a "bridge," we can use the law of sines.

We set up the ratios as shown above. We then cross multiply to eliminate the fractions. We then try to get x alone by dividing by sin69. We plug in the right side to our calculator and thus get 93.68 as our answer.

2. Law of Cosines
        Hannah has appointments in two different places. She can't be in two places at once, or CAN SHE? She asks Justin Timberlake if she can borrow his cloning machine. She then clones herself and orders her Hannah duplicate to go due east to Bakersfield. Hannah herself goes northeast to Riverside. The angle they create is 046*. The Hannah clone travels 25 miles to Bakersfield. Hannah travels 50 miles to Riverside. How far are the two apart now?

 We label everything accordingly. Since the problem said for the clone to go due easy, we know that Bakersfield is directly east of the starting line. The clone traveled 25 miles while Hannah traveled 50, so we label those. We also label the angle as 46*. Now that we have SAS, we can use the law of cosines.

We plug in our numbers into the formula as so. Let us substitute the variable a for x so that we can use b and c for the other sides. The formula is thus, to put it variably, a^2=b^2+c^2-2(b)(c)cosA. We then get our value when we plug it all into the calculator (do it all in one step) and get our next value. We have to square the value due to a being squared, so then we get our answer of 37.26 (miles).