At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 16 knots and ship B is sailing north at 22 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.)

Why is D^2=43880

Thank you very much for your help. everything makes sense and I got 100% on my answer.

To find the rate at which the distance between the two ships is changing, we can use the concept of relative velocity.

Let's consider the situation at 7 PM. From noon to 7 PM, there is a time difference of 7 hours.

Ship A is sailing west at 16 knots, so its position at 7 PM would be 16 knots * 7 hours = 112 nautical miles due west of its starting position at noon.

Ship B is sailing north at 22 knots, so its position at 7 PM would be 22 knots * 7 hours = 154 nautical miles north of its starting position at noon.

Now, let's draw a diagram to visualize the situation:

B
|
Ship A | Distance between ships
------(x)-----
|
|

Now, we want to find the rate at which the distance between the two ships is changing, which is essentially the rate of change of the length of the line connecting the two ships (labeled as "Distance between ships" in the diagram).

To find this, we can differentiate the equation of the distance between the ships with respect to time.

Using the Pythagorean theorem, we can see that:

Distance between ships = √(x^2 + 30^2)

Differentiating both sides of the equation with respect to time:

(d/dt)(Distance between ships) = (d/dt)√(x^2 + 900)

Now, let's find the value of x at 7 PM using the positions of the ships we calculated earlier:

x = 112 nautical miles

Substituting this into our equation:

(d/dt)(Distance between ships) = (d/dt)√(112^2 + 900)

To find the derivative, we can use the chain rule:

(d/dt)(Distance between ships) = (1/2√(112^2 + 900)) * (2x(dx/dt))

Now, we know that dx/dt is the velocity of ship A. Since ship A is sailing west at a constant speed of 16 knots, dx/dt = -16 knots (negative because it is moving west).

Substituting the values into the equation:

(d/dt)(Distance between ships) = (1/2√(112^2 + 900)) * (2 * 112 * -16)

Simplifying:

(d/dt)(Distance between ships) = (-16 * 112) / √(112^2 + 900)

Calculating this expression:

(d/dt)(Distance between ships) ≈ -56.812 knots

Therefore, the distance between the ships is changing at a rate of approximately 56.812 knots at 7 PM.

Did you make a diagram?

Let the time passed since noon be t hours
so the distance covered by ship B since then is 22t, and the distance covered by A since noon is 16t
Let the distance between the ships be D
I see a right-angled triangle where
D^2 = (16t+30)^2 + (22t)^2
2D(dD/dt) = 2(16t+30)(16) + 2(22t)(22)
dD/dt = (740t+480)/D

at 7:00 pm, t = 7 and
D^2 = 43880
D = √43880 = 209.476
and
dD/dt = (740(7)+480)/√43880 = 27.02 knots

check my arithmetic, I tend to make errors so early in the morning before my third cup of coffee.

The answer is correct. Thank u for ur help.