Answer:
30.25 mph ≈ 26.29 knots
Explanation:
You want to know the rate of separation after 6 hours of Ship A going west at 23 mph from a starting point 40 miles west of Ship B, which is going north at 20 mph.
Approach
There are some different ways this problem can be approached. We can write an expression for the distance between the ships, then find its derivative. We can describe the direction of one ship from the other as a vector, and find the component of the separation speed vector in that direction.
Distance expression
The distance between the two ships can be found using the distance formula.
The location of ship A can be described as (x, y) = (-40-23t, 0).
The location of ship B can be described as (x, y) = (0, 20t).
Then the distance formula tells us their separation is ...
d = √((0 -(-40 -23t))² +(20t -0)²) = √(929t² +1840t +1600)
The derivative of this is ...
d' = (1858t +1840)/(2√(929t² +1840t +1600)) = (929t +920)/d
At t=6, we have ...
d'(6) = (929(6)+920)/√((929(6) +1840)(6) +1600) = 6494/√46084
d'(6) ≈ 30.2508 . . . . miles per hour
You want the separation speed in knots, so this becomes ...
d' = (30.2508 mi/h)(0.8689762 nautical mi/mi) ≈ 26.29 knots
The distance between the ships is increasing at about 26.29 knots.
Vectors
The vector from the point of origin to Ship A after 6 hours is ...
a = (-40 -23(6), 0) = (-178, 0)
The vector from the point of origin to Ship B after 6 hours is ...
b = (0, 20(6) = (0, 120)
The direction of A from B is ...
a -b = (-178, 0) -(0, 120) = (-178, -120)
The speed of separation is likewise found to be
s = (-23, -20)
Then the "speed made good" in the direction between the ships is the dot product of the speed and the unit direction vector.
(s • (a -b))/|a -b| = (-23(-178) +(-20)(-120))/(√((-178)² +(-120)²)
= 6494/√46084 ≈ 30.2508 . . . . mi/h
You will notice a resemblance to the calculation above.
At 6 pm, the speed of separation is about 30.25 mph or 26.29 knots.
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Additional comment
We like to use available technology to help solve problems like this. Two different graphing calculator approaches are shown in the attachments.
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