At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 30 km/h. How fast is the distance between the ships changing at 4:00 PM?
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To find the rate at which the distance between the ships is changing at 4:00 PM, we can use the concept of derivatives from calculus.
Let's first establish a coordinate system: take ship B's initial position as the point (0,0). Since ship A is 150 km west of ship B, we can represent its initial position as (-150,0) in this coordinate system.
The position of ship A at time t can be given by the coordinates (35t - 150, 0) since it is sailing east at a constant rate of 35 km/h.
Similarly, the position of ship B at time t can be given by the coordinates (0, 30t) as it is sailing north at a constant rate of 30 km/h.
Now, the distance between the two ships can be found using the distance formula:
Distance^2 = (x2 - x1)^2 + (y2 - y1)^2
Using the coordinates of ship A and ship B at time t, the distance between them can be represented as:
Distance^2 = (35t - 150)^2 + (30t)^2
Now, to find how the distance is changing with respect to time, we need to differentiate this equation with respect to t.
Differentiating both sides, we get:
2 * Distance * d(Distance)/dt = 2 * (35t - 150) * 35 + 2 * (30t) * 30
Simplifying, we have:
Distance * d(Distance)/dt = (35t - 150) * 35 + (30t) * 30
Now, substitute the value of t as 4 (since we want to find the rate of change at 4:00 PM):
Distance * d(Distance)/dt = (35 * 4 - 150) * 35 + (30 * 4) * 30
Solving this equation will give us the rate at which the distance between the ships is changing at 4:00 PM.