To determine how rapidly the distance between the two ships is changing, we'll use the derivative.
Let's start by drawing a diagram to visualize the situation. We have ship A and ship B, with A initially 60 km due north of B.
A
|
|
60 km |
--------- B
Let's assume that at time t = 0, ship A is at the point (0, 60) and ship B is at the point (0, 0).
Since ship A is sailing east at 12 km/hr, after 2 hours, it will have traveled a horizontal distance of 12 km/hr * 2 hr = 24 km. Therefore, after 2 hours, ship A will be at the point (24, 60).
Since ship B is sailing north at 9 km/hr, after 2 hours, it will have traveled a vertical distance of 9 km/hr * 2 hr = 18 km. Therefore, after 2 hours, ship B will be at the point (0, 18).
B
|
|
18 km |
--------- A
|
|
|
|
| 24 km
Now, let's calculate the distance between ship A and ship B after 2 hours. We can use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Here, (x1, y1) = (0, 18) and (x2, y2) = (24, 60).
distance = sqrt((24 - 0)^2 + (60 - 18)^2)
= sqrt(24^2 + 42^2)
= sqrt(576 + 1764)
= sqrt(2340)
= 48.39 km (approximately)
Now, let's find out how rapidly the distance between the ships is changing. We want to find d(distance)/dt, the rate of change of the distance with respect to time.
To do this, we'll consider ship A and ship B as moving points. We know that ship A is moving horizontally at a rate of 12 km/hr, and ship B is moving vertically at a rate of 9 km/hr.
Let x be the horizontal distance between the ships, and y be the vertical distance between them. The distance between the ships can be calculated as:
distance = sqrt(x^2 + y^2)
To find the rate of change with respect to time, we'll differentiate the equation with respect to t:
d(distance)/dt = d(sqrt(x^2 + y^2))/dt
Using the chain rule, the equation becomes:
d(distance)/dt = (d(sqrt(x^2 + y^2))/dx)*(dx/dt) + (d(sqrt(x^2 + y^2))/dy)*(dy/dt)
Using the fact that dx/dt = 12 km/hr and dy/dt = 9 km/hr, we get:
d(distance)/dt = (x/sqrt(x^2 + y^2)) * 12 + (y/sqrt(x^2 + y^2)) * 9
Now we substitute the values we have at t = 2 hours:
x = 24 km (horizontal distance)
y = 42 km (vertical distance)
d(distance)/dt = (24/sqrt(24^2 + 42^2)) * 12 + (42/sqrt(24^2 + 42^2)) * 9
Simplifying this expression, we get:
d(distance)/dt = (24/48.39) * 12 + (42/48.39) * 9
d(distance)/dt = 12.47 km/hr
Therefore, the distance between the two ships is changing at a rate of approximately 12.47 km/hr after 2 hours.
Since the distance is increasing (because the two ships are moving away from each other), the rate is positive.