At noon, ship A is 50 miles north of ship B and is headed south at 16mph. Ship B is headed west at 12 mph. At what time are they closest together, and what is the minimum distance between them?

To find out when the two ships are closest together and the minimum distance between them, we can use the concepts of relative motion and the formula for distance.

First, let's establish a coordinate system. Let the initial position of ship B be the origin (0,0) on a graph. Ship A starts at (0,50) since ship A is 50 miles north of ship B.

The position of ship A (Ax, Ay) at time t can be calculated using the formula:
Ax = 0 (since it is always at the same longitude)
Ay = 50 - 16t (since it is moving south at 16 mph)

The position of ship B (Bx, By) at time t can be calculated using the formula:
Bx = -12t (since it is moving west at 12 mph)
By = 0 (since it is always at the same latitude)

Now, let's find the time at which ships A and B are closest together. We want to minimize the distance between them, which can be calculated using the distance formula:
d = sqrt((Ax - Bx)^2 + (Ay - By)^2)

Substituting the positions of ship A and ship B, we have:
d = sqrt((0 - (-12t))^2 + (50 - 16t - 0)^2)

Simplifying, we have:
d = sqrt((-12t)^2 + (50 - 16t)^2)
d = sqrt(144t^2 + 2500 - 1600t + 256t^2)
d = sqrt(400t^2 - 1600t + 2500)

To find the minimum distance, we can take the derivative of d with respect to t, set it to zero, and solve for t. The time at which the derivative is zero corresponds to the minimum distance.

Let's differentiate d with respect to t:
d' = (1/2) * (400t^2 - 1600t + 2500)^(-1/2) * (800t - 1600)
d' = (800t - 1600) / sqrt(400t^2 - 1600t + 2500)

Setting d' equal to zero and solving for t:
800t - 1600 = 0
800t = 1600
t = 1600 / 800
t = 2

So, the two ships are closest together at t = 2 hours.

Finally, let's calculate the minimum distance between the two ships at t = 2 hours:
d = sqrt(400(2)^2 - 1600(2) + 2500)
d = sqrt(1600 - 3200 + 2500)
d = sqrt(900)
d = 30 miles

Therefore, at t = 2 hours, the two ships are closest together, and the minimum distance between them is 30 miles.