At noon, ship A is 180 km west of ship B. Ship A is sailing east at 40 km/h and ship B is sailing north at 35 km/h. How fast is the distance between the ships changing at 4:00 PM?

≈ 29 km/h

Step-by-step explanation:

If t is the number of hours since noon, then the distance between ship A and the origin is 180 − 40t, and the distance between ship B and the origin is 35t.

Using Pythagorean theorem, the distance between the ships is:

d² = (35t)² + (180 − 40t)²

d² = 1225t² + 32400 − 14400t + 1600t²

d² = 2825t² − 14400t + 32400

Taking derivative with respect to time:

2d dd/dt = 5650t − 14400

dd/dt = (2825t − 7200) / d

First, find d when t = 4.

d² = 2825 (4)² − 14400 (4) + 32400

d² = 20000

d = 100√2

Now find dd/dt.

dd/dt = (2825(4) − 7200) / (100√2)

dd/dt = 41/√2

dd/dt ≈ 29 km/h

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