At noon, ship A is 150 km west of ship B. Ship A is sailing east at 25 km/h and ship B is sailing north at 20 km/h. How fast is the distance between the ships changing at 4:00 PM?
To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity.
First, let's find the positions of the ships at 4:00 PM. We know that ship A is sailing east at a constant speed of 25 km/h, so it would have traveled for 4 hours starting from noon. Therefore, the position of ship A at 4:00 PM would be 25 km/h * 4 h = 100 km east of its initial position.
Ship B, on the other hand, is sailing north at a constant speed of 20 km/h. Since it started 150 km west of ship A at noon, it would have traveled for 4 hours as well. Therefore, the position of ship B at 4:00 PM would be 20 km/h * 4 h = 80 km north of its initial position.
Now, we can draw a triangle with ship A, ship B, and the distance between them as the sides. The horizontal side represents the position of ship A (100 km east), the vertical side represents the position of ship B (80 km north), and the hypotenuse represents the distance between the two ships.
By using the Pythagorean theorem, we can calculate the distance between the ships at 4:00 PM. The distance (D) can be found using the equation:
D^2 = (100 km)^2 + (80 km)^2
Solving this equation, we find that D ≈ 128.06 km.
To find the rate at which the distance between the ships is changing, we need to differentiate the equation with respect to time (t). However, we need to find the instantaneous rate of change, so we will use the Chain Rule to differentiate both sides of the equation. The Chain Rule states that d(D^2)/dt = 2D * dD/dt.
Differentiating the equation, we get:
2D * dD/dt = 2 * 100 km * (dD/dt)
Now, let's find dD/dt, the rate at which the distance is changing.
Since ship A is traveling horizontally, the rate of change of its position with respect to time (dx/dt) is 25 km/h. Therefore, dD/dt = dx/dt.
So, dD/dt = 25 km/h.
Now, substitution into the previous equation, we get:
2D * dD/dt = 2 * 100 km * (25 km/h) = 5000 km^2/h.
This means that at 4:00 PM, the distance between the ships is changing at a rate of 5000 km^2/h.