Final answer:
To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity and the Pythagorean theorem. By differentiating the equation and substituting the given values, we can determine how rapidly the distance between the ships is changing 4 hours later.
Step-by-step explanation:
To find the rate at which the distance between the ships is changing, we need to use the concept of relative velocity.
First, let's find the velocity of each ship:
Ship A sails east at a speed of 15 miles per hour, so its velocity is 15 miles per hour due east.
Ship B sails north at a speed of 28 miles per hour, so its velocity is 28 miles per hour due north.
To find the rate at which the distance is changing, we can use the Pythagorean theorem:
Distance^2 = (Change in x)^2 + (Change in y)^2
Let's say the distance between the ships at noon is represented by D:
D^2 = (60 + 15t)^2 + (28t)^2
where t is the time in hours after noon. We can differentiate the equation with respect to time:
2D(dD/dt) = (2)(15)(15t + 60) + (2)(28)(28t)
Simplifying the equation, we get:
dD/dt = (15)(15t + 60) + (28)(28t) / 2D
Substituting t = 4 into the equation and solving, we can find the rate at which the distance between the ships is changing 4 hours later.