Related Rates
I am having trouble finding the equation to use to fit this all together.
At noon, ship A is 100km west of ship B. Ship A is sailing south at 35 km/s and ship B is sailing north at 25 km/s. How fast is the distance between the ships changing at 4:00pm.
To solve this problem using related rates, we need to find an equation that relates the variables in the problem and differentiate it with respect to time to find the rate of change.
Let's consider the situation at 4:00 pm. Let D represent the distance between the ships, and let t represent time in hours since noon. At noon, D = 100 km since Ship A is 100 km west of Ship B.
Since Ship A is sailing south at 35 km/h, the distance traveled by Ship A from noon to 4:00 pm (in 4 hours) is 35 km/h × 4 h = 140 km. Hence, at 4:00 pm, Ship A is at a location 140 km south of the position at noon. Therefore, the position of Ship A at 4:00 pm is D + 140 km.
Similarly, Ship B is sailing north at 25 km/h, and the distance traveled by Ship B from noon to 4:00 pm (in 4 hours) is 25 km/h × 4 h = 100 km. Hence, at 4:00 pm, Ship B is at a location 100 km north of the position at noon. Therefore, the position of Ship B at 4:00 pm is D + 100 km.
Now, let's find the equation that relates the variables. At 4:00 pm, the distance between the ships (D) is the difference between their positions:
D = (D + 140 km) - (D + 100 km) = D + 140 km - D - 100 km = 40 km.
Now, we can differentiate the equation with respect to time (t), using the chain rule:
dD/dt = d(D + 140 km - D - 100 km)/dt.
Since the D terms cancel out, we have:
dD/dt = 140 km/h - 100 km/h = 40 km/h.
Therefore, the rate at which the distance between the ships is changing at 4:00 pm is 40 km/h.