Derivative confusion

Question

At noon, ship A is 180 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM? 

Answer 1

Hi Ju R., this is an example of an instantaneous rate and Pythagorean theorem problem. What you need to do is visualize the right triangle based on Ship A and Ship B's movement. Ship A is sailing east (forming one side of the right triangle), and Ship B is sailing north (forming the other side). The distance between the ships will be represented by the hypotenuse, in the equation c²=a²+b² , but since a and b are dependent on time (t=# of hours), its easy to rearrange this as c²=(180-35t)² + (25t)² which simplifies to c =sqrt [(180-35t)² + (25t)²]. Now, just use the chain rule to take the derivative relative to time and plug in 4 (since noon --> 4 pm). Solving for dc/dt will give you instantaneous rate of change of distance between them at 4pm. Hope this helps!