At noon, ship A is 10 nautical miles due west of ship B. Ship A is sailing west at 24 knots and ship B is sailing north at 19 knots. How fast (in knots) is the distance between the ships changing at 6 PM?
A variation of an old standard
Just change the numbers to yours
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To find the rate at which the distance between the two ships is changing at 6 PM, we need to determine the speed at which the distance is increasing or decreasing with respect to time. We can then use the Pythagorean theorem and trigonometry to solve the problem.
Let's break down the problem step by step:
1. Convert the given speeds to knautical miles per hour (knots):
- Ship A is sailing at 24 knots.
- Ship B is sailing at 19 knots.
2. Determine the time elapsed from noon to 6 PM: 6 PM - 12 PM = 6 hours.
3. Calculate the distance covered by each ship from noon to 6 PM:
- Ship A: 24 knots * 6 hours = 144 nautical miles due west.
- Ship B: 19 knots * 6 hours = 114 nautical miles due north.
4. Use the Pythagorean theorem to find the distance between the two ships at 6 PM:
- Distance^2 = (10 nautical miles + 144 nautical miles)^2 + (0 nautical miles + 114 nautical miles)^2.
- Distance^2 = 154^2 + 114^2.
*Note: Since the ships are moving on right angles, we can add the distances covered by each ship in a right-angled triangle.*
5. Calculate the square root of the sum of the squares to find the distance between the two ships at 6 PM:
- Distance ≈ √(154^2 + 114^2).
6. Differentiate the equation of distance with respect to time to find the rate of change of the distance between the ships at 6 PM.
By following these steps, you can calculate the speed (in knots) at which the distance between the ships is changing at 6 PM.