SOLUTION: two cars, a and b, start from the same point and the same time and travel along straight roads that form 60 degrees with each other. after two hours, car A has traveled 120 km and

Algebra ->  Trigonometry-basics -> SOLUTION: two cars, a and b, start from the same point and the same time and travel along straight roads that form 60 degrees with each other. after two hours, car A has traveled 120 km and       Log On


   

Question 623927: two cars, a and b, start from the same point and the same time and travel along straight roads that form 60 degrees with each other. after two hours, car A has traveled 120 km and car B traveled 100 km. how far apart are the cars after two hours?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
It may help to draw a diagram:
  • Draw a triangle with a 60 degree angle. (It doesn't have to be exact.) This triangle may or may not be a right triangle. We do now know yet.
  • Label the vertex of the 60 degree angle as "C".
  • Label the other two verices as "A" and "B".
  • Vertex C is where the cars started. Vertex B is where bar B is after the two hours. And vertex A is where car A is after the two hours.
  • Since the cars have been traveling for two hours and since car A travels at 120 kph and car B travels at 100 kph, they have traveled 240 km and 200 km, respectively. Label side CA as 240 km and side CB as 200 km.
Our task is to find how far apart car A and car B are after the two hours. In terms of the triangle, we are looking for the length of side AB.

Knowing and angle and two sides of a triangle (which may or may not be a right triangle) and wanting to find the third side is a job for the Law of Cosines:
c%5E2+=+a%5E2%2Bb%5E2-2ab%2Acos%28C%29
where "c" is the side opposite to angle C. The angle we know is the angle opposite the side we want to find. So we can insert our numbers into the formula as follows:
c%5E2+=+%28240%29%5E2+%2B+%28200%29%5E2+-+2%28240%29%28200%29%2Acos%2860%29
Since 60 is s special angle, we should know that cos(60) = 0.5 (or 1/2). So now our formula is:
c%5E2+=+%28240%29%5E2+%2B+%28200%29%5E2+-+2%28240%29%28200%29%2A%280.5%29
Now we solve for c. First we simplify:
c%5E2+=+57600+%2B+40000+-+2%28240%29%28200%29%2A%280.5%29
c%5E2+=+57600+%2B+40000+-+48000
c%5E2+=+49600
Then we find the square root of each side. (We will ignore the negative square root since "c" is a distance which should not be considered negative.).
c+=+sqrt%2849600%29
Simplifying the square root...
c+=+sqrt%28100%2A496%29
c+=+sqrt%28100%2A4%2A124%29
c+=+sqrt%28100%2A4%2A4%2A31%29
c+=+sqrt%28100%29%2Asqrt%284%29%2Asqrt%284%29%2Asqrt%2831%29
c+=+10%2A2%2A2%2Asqrt%2831%29
c+=+40%2Asqrt%2831%29
This is an exact expression for the distance between cars A and B after the two hours. If you want/need a decimal approximation for the answer, use your calculator on the square root and then multiply it by 40.