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?
<pre>Let speed of 1<sup>st</sup> boat be S
Then we get the following: {{{matrix(1,3, 18/S, "=", 2/sqrt(3))}}}
{{{matrix(1,3, 2S, "=", 18sqrt(3))}}} ------- Cross-multiplying
S, or speed of 1<sup>st</sup> boat = {{{matrix(1,3, 18sqrt(3)/2, "=", 9sqrt(3))}}}
We then get: 	Distance traveled by 1<sup>st</sup> boat in 5 hours: 
               {{{matrix(1,4, 5(9)sqrt(3), "=", 45sqrt(3), km)}}}
Distance traveled by 2<sup>nd</sup> boat in 5 hours: 5(18) = 90 km

The side opposite the 30<sup>o</sup> 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 30<sup>o</sup> angle (A) be a
Then we get: {{{matrix(1,3, a^2, "=", b^2 + c^2 - 2bc Cos (A))}}}  
             {{{matrix(1,3, a^2, "=", (45*sqrt(3))^2 + 90^2 - 2(45*sqrt(3))(90)Cos (30^o))}}}        
             {{{matrix(1,3, a^2, "=", 3(45^2) + 90^2 - 180(45sqrt(3)) * (sqrt(3)/2))}}}       
             {{{matrix(1,3, a^2, "=", 3(45)^2 + 90^2 - (180 * 45 * 3)/2)}}} 
             {{{matrix(1,3, a^2, "=", 3(45)^2 + 90^2 - (90cross(180) * 45 * 3)/cross(2))}}}
             {{{matrix(1,3, a^2, "=", 3(45)^2 + 90^2 - 90(45)(3))}}}
             {{{matrix(1,3, a^2, "=", "2,025")}}}
             Distance the boats are apart, or {{{highlight_green(matrix(1,6,              a, "=", sqrt("2,025"), "=", 45, km))}}}
             OR
             {{{matrix(1,3, a^2, "=", b^2 + c^2 - 2bc Cos (A))}}}  
             {{{matrix(1,3, a^2, "=", (45*sqrt(3))^2 + 90^2 - 2(45*sqrt(3))(90)Cos (30^o))}}}
             {{{matrix(1,3, a, "=", sqrt((45*sqrt(3))^2 + 90^2 - 2(45*sqrt(3))(90)Cos (30^o)))}}}
             Distance the boats are apart, or {{{highlight_green(matrix(1,6, a, "=", sqrt("2,025"), "=", 45, km))}}}