At noon, ship A is 10 nautical miles due west of ship B. Ship A is sailing west at 16 knots and ship B is sailing north at 24 knots. How fast (in knots) is the distance between the ships changing at 3 PM?
This is what I got but it's not right 28.727
I also got your same answer even correct to the last decimal place
Hmm. Stupid online hw wants it perfectly I guess. Thanks though.
Can you explain how you got your answer?
To find the speed at which the distance between the ships is changing, we can use the concept of relative velocity. Here's how you can solve it:
1. Start by drawing a diagram to represent the scenario. Place ship A at the origin (0,0) and ship B 10 nautical miles due east (10,0).
2. Ship A is sailing west at a constant speed of 16 knots. This means its velocity vector can be represented as (-16,0) knots.
3. Ship B is sailing north at a constant speed of 24 knots. Therefore, its velocity vector can be represented as (0,24) knots.
4. Now, we can find the distance between the two ships at any given time. At noon, the distance between the ships is equal to the magnitude of the vector from A to B, which is the distance between their positions. In this case, it is 10 nautical miles.
5. We need to find how this distance changes over time. To do this, we can calculate the time derivative of the distance at a given time.
6. By using the chain rule of differentiation, differentiate the equation (distance)^2 = (x_B - x_A)^2 + (y_B - y_A)^2 with respect to time.
d(distance^2) / dt = 2 * (x_B - x_A) * (dx_B/dt) + 2 * (y_B - y_A) * (dy_B/dt)
Substituting the given values, we have:
d(distance^2) / dt = 2 * (10) * (0) + 2 * (0) * (24) = 0
Therefore, the rate at which the distance between the ships is changing is 0 knots.