document.write( "Question 1192807: A ship is travelling north at a constant speed of 40 km/h. A second ship, which is initially 60 km to the east of the first ship, is travelling northwest at a speed of 50 km/h. What is the minimum distance between these two ships? \n" ); document.write( "

Algebra.Com's Answer #824740 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Let the starting position of the first ship be (0,0).

\n" ); document.write( "Then from the given information, the position of the first ship after t hours is ( $\"0\"$ , $\"40t\"$ ).

\n" ); document.write( "The initial position of the second ship is (60,0); it is moving 50 km/h in the northwest direction, which is $\"-25sqrt%282%29\"$ km/h in the x direction and $\"25sqrt%282%29\"$ km/h in the y direction.

\n" ); document.write( "So the position of the second ship after t hours is ( $\"60-%2825sqrt%282%29%29t\"$ , $\"25sqrt%282%29t\"$ ).

\n" ); document.write( "The distance between the two ships after t hours is then the distance between the points ( $\"0\"$ , $\"40t\"$ ) and ( $\"60-%2825sqrt%282%29%29t\"$ , $\"25sqrt%282%29t\"$ ). That distance, by the distance formula (AKA Pythagorean Theorem) is

\n" ); document.write( " $\"sqrt%28%2860-%2825sqrt%282%29%29t%29%5E2%2B%2840t-25sqrt%282%29t%29%5E2%29\"$

\n" ); document.write( "You could find the minimum distance between the two ships by finding where the derivative of that distance function is zero; but if is far easier to find the minimum value by graphing the function on a graphing calculator.

\n" ); document.write( "The answer I got using my TI-83 calculator was a minimum distance between the ships of about 3.65 km at t=1.69 hours.

\n" ); document.write( "ANSWER: about 3.65km

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