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Okie
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At noon, ship A is 10 nautical miles due west of ship B. Ship A is sailing west at 16 knots and ship B is sailing north at 19 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
Note: Draw yourself a diagram which shows where the ships are at noon and where they are "some time" later on. You will need to use geometry to work out a formula which tells you how far apart the ships are at time t, and you will need to use "distance = velocity * time" to work out how far the ships have traveled after time t.
So the way that I did (but it is wrong)
at 4pm 4 hours have past so ship A has traveled 16*4 = 64 knots and ship B has traveled 19*4 = 76 knots.
by pythagoras the distance between them at this time is "r"
r² = 64² + 76²
r² = 9872
r = 4√617
Now:
at any time if A is how far ship A has traveled and B is how far ship B has traveled then
r² = A² + B²
differentiating with respect to time
2r dr/dt = 2A dA/dt + 2B dB/dt
r dr/dt = A dA/dt + B dB/dt
we know dA/dt = 16 knots, dB/dt = 19 knots
at 4 pm
4√617 dr/dt = 64*16 + 76*19
4√617 dr/dt = 2468
dr/dt = 2468/4√617
dr/dt = 617/√617
dr/dt = √617 knots
so at 4PM the distance between them is increasing by 24.84 knots (2dp)
Note: Draw yourself a diagram which shows where the ships are at noon and where they are "some time" later on. You will need to use geometry to work out a formula which tells you how far apart the ships are at time t, and you will need to use "distance = velocity * time" to work out how far the ships have traveled after time t.
So the way that I did (but it is wrong)
at 4pm 4 hours have past so ship A has traveled 16*4 = 64 knots and ship B has traveled 19*4 = 76 knots.
by pythagoras the distance between them at this time is "r"
r² = 64² + 76²
r² = 9872
r = 4√617
Now:
at any time if A is how far ship A has traveled and B is how far ship B has traveled then
r² = A² + B²
differentiating with respect to time
2r dr/dt = 2A dA/dt + 2B dB/dt
r dr/dt = A dA/dt + B dB/dt
we know dA/dt = 16 knots, dB/dt = 19 knots
at 4 pm
4√617 dr/dt = 64*16 + 76*19
4√617 dr/dt = 2468
dr/dt = 2468/4√617
dr/dt = 617/√617
dr/dt = √617 knots
so at 4PM the distance between them is increasing by 24.84 knots (2dp)