Final answer:
The distance between HMS Dreadnought and USS Mona Lisa is initially increasing at a rate of 50 miles per hour, calculated using the derivative of the Pythagorean theorem to find the rate of change in distance.
Step-by-step explanation:
To determine how quickly the distance between the HMS Dreadnought and USS Mona Lisa is increasing, we can use the concept of relative velocity in two dimensions.
Let's assume Montauk is the origin of our coordinate system.
At a certain point in time, the HMS Dreadnought will be at the position (0, -40 - 20t) where t is the time in hours since it was 40 miles south of Montauk, and the USS Mona Lisa will be at the position (50 + 30t, 0) since it heads east at 30 mph from a point 50 miles east of Montauk.
The distance d between the two ships is given by the Pythagorean theorem: d = √((50 + 30t)^2 + (-40 - 20t)^2). The rate at which this distance increases is the derivative of the distance with respect to time, d'/dt.
To find d'/dt, we differentiate using the chain rule:
d'/dt = d((50 + 30t)^2 + (-40 - 20t)^2)/dt × d/d((50 + 30t)^2 + (-40 - 20t)^2).
Working through the differentiation and simplification,
we get d'/dt = (2×50×30 + 2×30^2t + 2×(-40)×(-20) + 2×(-20)^2t) / (2×d), which simplifies further to d'/dt = (60×50 + 120×t - 80×40 - 40×t) / d.
The final answer in 20 words or so: At the initial position (t=0), the distance is increasing at a rate of approximately 50 miles per hour.
Please mention the correct option in the final answer: The initial rate at which the distance between the ships is increasing is 50 miles per hour.