SOLUTION: Two cargo boats leave the same port at the same time. The direction of one boat is at an angle of 30 degrees to the other. If the ratio of the speed of the first boat to the second

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Question 1125459: Two cargo boats leave the same port at the same time. The direction of one boat is at an angle of 30 degrees to the other. If the ratio of the speed of the first boat to the second is sqrt 3 over 2, and if the second boat travels at 18km/h, how far apart, in km, are they at the end of 5 hours?
Found 2 solutions by ankor@dixie-net.com, MathTherapy:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Two cargo boats leave the same port at the same time.
The direction of one boat is at an angle of 30 degrees to the other.
If the ratio of the speed of the first boat to the second is sqrt 3 over 2, and if the second boat travels at 18km/h, how far apart, in km, are they at the end of 5 hours?
:
Find the speed of the 1st boat (s)
s%2F18 = sqrt%283%29%2F2
cross multiply
2s+=+18%2Asqrt%283%29
s = 31.177%2F2
s = 15.6 km/h is the speed of the 2nd boat
:
Find the distance each has traveled in 5 hrs
5*15.6 = 78 km traveled by the 1st boat
5*18 = 90 km by the 2nd boat
:
The distance between the boat is the side (a) opposite the 30 degree angle (A)
Use the law of cosines: a^2 = b^2 + c2 - 2bc*Cos(a),where
b = 78
c = 90
a%5E2+=+78%5E2+%2B+90%5E2+-+2%2878%2A90%29%2ACos%2830%29
a^2 = 6084 + 8100 - 2(7020)*.866
Do the math
a = sqrt%282025%29
a = 45 km apart in 5 hrs

Answer by MathTherapy(10549) About Me  (Show Source):
You can put this solution on YOUR website!
Two cargo boats leave the same port at the same time. The direction of one boat is at an angle of 30 degrees to the other. If the ratio of the speed of the first boat to the second is sqrt 3 over 2, and if the second boat travels at 18km/h, how far apart, in km, are they at the end of 5 hours?
Let speed of 1st boat be S

Then we get the following: matrix%281%2C3%2C+18%2FS%2C+%22=%22%2C+2%2Fsqrt%283%29%29

matrix%281%2C3%2C+2S%2C+%22=%22%2C+18sqrt%283%29%29 ------- Cross-multiplying

S, or speed of 1st boat = matrix%281%2C3%2C+18sqrt%283%29%2F2%2C+%22=%22%2C+9sqrt%283%29%29

We then get: 	Distance traveled by 1st boat in 5 hours: 

               matrix%281%2C4%2C+5%289%29sqrt%283%29%2C+%22=%22%2C+45sqrt%283%29%2C+km%29

Distance traveled by 2nd boat in 5 hours: 5(18) = 90 km



The side opposite the 30o angle represents the distance between the boats after 5 hours

Therefore, with 2 SIDES and an INCLUDED ANGLE (SAS), we use the law of cosines.

Let the side opposite the 30o angle (A) be a

Then we get: matrix%281%2C3%2C+a%5E2%2C+%22=%22%2C+b%5E2+%2B+c%5E2+-+2bc+Cos+%28A%29%29  

                     

                    

              

             

             matrix%281%2C3%2C+a%5E2%2C+%22=%22%2C+3%2845%29%5E2+%2B+90%5E2+-+90%2845%29%283%29%29

             matrix%281%2C3%2C+a%5E2%2C+%22=%22%2C+%222%2C025%22%29

             Distance the boats are apart, or 

             OR

             matrix%281%2C3%2C+a%5E2%2C+%22=%22%2C+b%5E2+%2B+c%5E2+-+2bc+Cos+%28A%29%29  

             

             

             Distance the boats are apart, or