SOLUTION: Ship A was sailing due to South at the rate of 6 miles per hour and another ship B was sailing due to East at the rate of 8 miles per hour. At 4 PM ship B crossed the position wher

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Question 826008: Ship A was sailing due to South at the rate of 6 miles per hour and another ship B was sailing due to East at the rate of 8 miles per hour. At 4 PM ship B crossed the position where hip A was 2 hours before.
1) At what rate were they approaching or separating at 3 PM ?
2) At what rate were they changing the distance between them at 5 PM ?
3) When was the distance between them not changing ?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Ship A was sailing due to South at the rate of 6 miles per hour and another ship B was sailing due to East at the rate of 8 miles per hour. At 4 PM ship B crossed the position where hip A was 2 hours before.
Use 2 PM as the starting point.
Ship A is 12 miles North of it, and ship B is at the point.
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Find the distance between the ships as a function of time (in hours).
t = 0 at the starting point.
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Ship A's distance from the SP is 12 - 6t miles
Ship B's distance from the SP is 8t miles
The distance between them is the hypotenuse of a right triangle.
s%5E2+=+%2812-6t%29%5E2+%2B+%288t%29%5E2+=+100t%5E2+-+144t+%2B+144
s+=+sqrt%28100t%5E2+-+144t+%2B+144%29
Find the relative speed, ds/dt
ds/dt = %281%2F2%29%2Fsqrt%28100t%5E2+-+144t+%2B+144%29%2A%28200t+-+144%29
s'(t) = %28100t-72%29%2Fsqrt%28100t%5E2+-+144t+%2B+144%29
s'(t) = %2850t-36%29%2Fsqrt%2825t%5E2+-+36t+%2B+36%29
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1) At what rate were they approaching or separating at 3 PM ?
3 PM is t = 1
s'(1) = %2850-36%29%2Fsqrt%2825+-+36+%2B+36%29
= 14/5 = 2.8 mi/hr
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2) At what rate were they changing the distance between them at 5 PM ?
5 PM --> t = 3
s'(3) = %2850%2A3-36%29%2Fsqrt%2825%2A9+-+36%2A3+%2B+36%29
= 114%2Fsqrt%28153%29
= 38%2Fsqrt%2817%29
=~ 9.216 mi/hr
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3) When was the distance between them not changing ?
s'(t) = %2850t-36%29%2Fsqrt%2825t%5E2+-+36t+%2B+36%29 = 0
50t-36 = 0
t = 0.72 hrs past 2 PM
t = 2:43:12 PM
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The speeds in parts 1 & 2 are both separating, since they're past the time in part 3.
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It appears to be a simple problem, but is complex. At least it's a right triangle.