ship nautical miles problem unsolveable

Question

At noon, ship A is 10 nautical miles due west of ship B. Ship A is sailing west at 15 knots and ship B is sailing north at 17 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Answer 1

when you solve out for d you get distance at 5 pm between the two ships = 120.21.

We then need to take the derivative of the distance formula with respect to time.

we know that the distance formula is x^2 = A^2 + B^2 where x = distance between A and B, A = location of ship A at 5 pm, and B = location of ship B at 5 pm

taking the derivative with respect to time (because this is a rate of change problem) we get: d/dt(x^2) = d/dt(A^2) + d/dt(B^2)

next line of work is actually performing the derivative: 2x(dx/dt) = 2A(dA/dt) + 2B(dB/dt) where dA/dt and dB/dt are the rates of movement in knots of the ships and dx/dt is the rate change in distance between the two ships, which is what we are solving for.

we can divide by 2 to make the equation simpler, which makes the new equation this:

x(dx/dt) = A(dA/dt) + B(dB/dt)

then we plug in what we have solved for so far, x = 120.21, A = 85 (positive since we are referring to miles), dA/dt = 15 knots, B = 85, and dB/dt = 17 knots

120.21(dx/dt) = 85(15) + 85(17)

simplify: dx/dt = 2720/120.21 = 22.63 nautical miles per hour