Boat A leaves the dock at 12pm heading due South at a constant speed of 20 km/h.

Meanwhile, at 12pm Boat B, a pirate ship, is 15 km due West of the dock where boat
A started, and is sailing due East at a constant speed of 15 km/h. If the pirates get
within 10 km of another boat, they will spot it and eventually capture it. Is Boat A
in danger of getting captured by the pirates on Boat B?

Make a diagram showing the ships at t hours after 12:00 pm.

Let D be the position of the dock and A and B the position of the boats.
then DA = 20t km
and BD = (15-15t) km

AB^2 = (20t)^2 + (15-15t)^2

let's find the minimum distance AB
2AB d(AB)/dt = 800t + 2(15-15t)(-15)
= 0 for a minimum of AD
800t - 30(15-15t) = 0
800t - 450 + 450t =
t = 450/1250
t = .36 hours
when t = .36
(AB)^2 = 51.84 + 92.16
AB = √144
= 12

Since they have to be within 10 km to be seen, they are ok.

To determine if Boat A is in danger of getting captured by Boat B, we need to analyze their respective positions and speeds.

First, let's consider the paths and speeds of both boats:

- Boat A is traveling due South at a constant speed of 20 km/h.
- Boat B is initially 15 km due West of the dock and is sailing due East at a constant speed of 15 km/h.

Now, let's break down the problem step by step:

1. Determine the time it takes for both boats to cover their respective distances:
- Since Boat A is traveling at a constant speed of 20 km/h, it will take 15 km / 20 km/h = 0.75 hours (or 45 minutes) to reach the longitude of Boat B.

2. Calculate the distance between the current position of Boat A and the longitude of Boat B at that time:
- After 45 minutes (0.75 hours), Boat A will reach the longitude of Boat B and will still be traveling South.
- The distance between their longitudes will be 15 km + Boat A's distance covered in 45 minutes.

3. Analyze if the distance between Boat A and Boat B is less than 10 km:
- If the distance calculated in step 2 is less than 10 km, Boat A is in danger of getting captured by Boat B since the pirates will spot it.

To further calculate the distance and determine the result, we need to know the distance covered by Boat A in the given timeframe. The problem description does not provide this information.