SOLUTION: Two botas starts at the same point. One sail due east starting 10 A.M. at a constant rate of 20 kph. The other sail due south starting 11 A.M. at constant rate of 9 kph. How fast a

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Question 1082937: Two botas starts at the same point. One sail due east starting 10 A.M. at a constant rate of 20 kph. The other sail due south starting 11 A.M. at constant rate of 9 kph. How fast are they separating at noon?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The distance between them is the hypotenuse of a right triangle who's legs are measured by the rate of each boat.
A%5E2%2BB%5E2=H%5E2
Let's measure time starting at 11 AM,
A=20t%2B20=20%28t%2B1%29
B=9t
So then,
400%28t%2B1%29%5E2%2B%289t%29%5E2=H%5E2
400%28t%5E2%2B2t%2B1%29%2B81t%5E2=H%5E2
H%5E2=481t%5E2%2B800t%2B400
So at noon, t=2,
H%5E2=481%284%29%2B800%282%29%2B400
H%5E2=1924%2B1600%2B400
H%5E2=3924
H=sqrt%283924%29
H=6sqrt%28109%29
So now to find the rate of separation, implicitly differentiate the equation above with respect to time.
2A%28dA%2Fdt%29%2B2B%28dB%2Fdt%29=2H%28dH%2Fdt%29
A=20%282%2B1%29=60
B=9%282%29=18
dA%2Fdt=20
dB%2Fdt=9
Substituting,
2%2860%29%2820%29%2B2%2818%29%289%29=2%286sqrt%28109%29%29%28dH%2Fdt%29
12sqrt%28109%29%28dH%2Fdt%29=2724%29
dH%2Fdt=227%2Fsqrt%28109%29
highlight%28dH%2Fdt=%28227%2F109%29sqrt%28109%29%29kph