one ship (A) is sailing due south at 16 miles per hour and a second ship (B), 32 miles south of A, is sailing due east at 12 miles per hour. At what rate are they approaching at the end of 1 hour?
-5.6 mph
To find the rate at which the two ships are approaching each other after one hour, we need to use the concept of vectors and vector addition.
Let's set up a coordinate system where the north direction is positive and the east direction is positive.
Ship A is sailing due south at a speed of 16 miles per hour. This means that its velocity vector can be represented as (-16, 0) miles per hour. The negative sign indicates that the velocity is in the opposite direction of the positive y-axis (north).
Ship B is sailing due east at a speed of 12 miles per hour. Its velocity vector can be represented as (0, 12) miles per hour.
To find the rate at which they are approaching each other, we need to add the two velocity vectors together.
Adding (-16, 0) and (0, 12) gives us (-16, 12). This means that after one hour, the ships are approaching each other at a rate of 16 miles per hour in the south direction and 12 miles per hour in the east direction.
To find the overall rate at which they are approaching, we can use the Pythagorean theorem to calculate the magnitude of the resulting vector. The magnitude of (-16, 12) is given by sqrt((-16)^2 + 12^2) = sqrt(256 + 144) = sqrt(400) = 20.
Therefore, at the end of one hour, the two ships are approaching each other at a rate of 20 miles per hour.