The first boat is sailing towards the North at the speed of 6 km per hour and sees that another boat is heading east at the speed of 5 km per hour. The second boat is to the northwest of the first boat. what is the minimum distance between two boats, assuming that they do not change their speed or course?
at t=0, let the 1st boat be at (0,0)
The 2nd boat is at (-c,c)
So, at time t, the 1st boat is at (0,6t)
and the 2nd boat is at (-c+5t,c)
The distance is thus
d^2 = (-c+5t)^2 + (c-6t)^2
= c^2 - 10ct + 25t^2 + c^2 - 12ct + 36t^2
= 2c^2 - 22ct + 51t^2
2d dd/dt = -22c + 102t
dd/dt = (51t-11c)/d
dd/dt=0 at t = 11c/51
So, plug that into the distance formula to get d. As you can see, the nearest distance depends on how far away the two boats were at first.
To find the minimum distance between the two boats, we can use the Pythagorean theorem. Let's assume that the first boat starts at the origin (0,0) and is moving towards the North. The second boat is to the northwest, which means it is at some point with both positive x and positive y coordinates.
The first boat is moving at a speed of 6 km/h towards the North, so after t hours, its position is (0, 6t). The second boat is moving towards the East at a speed of 5 km/h, so after t hours, its position is (5t, 5t).
Now, we can calculate the minimum distance between the two boats at any given time using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's substitute the positions of the two boats at time t:
Distance = sqrt((5t - 0)^2 + (5t - 6t)^2)
= sqrt((5t)^2 + (-t)^2)
= sqrt(25t^2 + t^2)
= sqrt(26t^2)
To find the minimum distance, we need to find the minimum value of t. Since both boats are moving at a constant speed, the minimum distance occurs when they have traveled the same amount of time, t.
Equating the distances traveled by the two boats:
6t = 5t
Solving for t:
t = 0
Therefore, the minimum distance between the two boats occurs when t = 0, which means the boats are at their starting positions. At this point, the distance between them is 0 km.
Hence, the minimum distance between the two boats is 0 km.