Calculus Help (answer with explanation please)

Question

At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 16 knots and ship B is sailing north at 16 knots. How fast (in knots) is the distance between the ships changing at 7 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Answer 1

Draw a sketch of the situation:

You can see that d₁ = 16t (where "t" is in hours) making d₃ = 16t + 50.

You can also see that d² = 16t as well.

So that make d₄ = √((16t + 50)² + (16t)²)

d₄(t) = √(256t² + 1600t + 2500 + 256t²) = √(512t² + 1600t + 2500)

To make things easier, lets define the distance between ships A and B as "x" instead of d₄ so:

x(t) = (512t² + 1600t + 2500)^1/2

T find the rate of change, take the derivative

so dx/dt = 1/2(512t² + 1600t + 2500)^-1/2(1024t + 1600) per the power rule and the chain rule.

At t = 7, we can find dx/dt by plugging in x = 7 yielding 22.26 nautical miles per hour