At noon, ship A is 30 nautical miles due west of ship B. Ship A is sailing west at 25 knots and ship B is sailing north at 25 knots. How fast (in knots) is the distance between the ships changing at 6 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
To find how fast the distance between the ships is changing at 6 PM, we need to calculate the rate of change of distance (or speed) between the ships.
Given information:
- Ship A is 30 nautical miles due west of ship B at noon.
- Ship A is sailing west at 25 knots.
- Ship B is sailing north at 25 knots.
Step 1: Find the distance traveled by each ship from noon to 6 PM.
Since ship A is sailing at a constant speed of 25 knots westward from noon to 6 PM, the distance traveled by ship A is 25 knots multiplied by 6 hours, which equals 150 nautical miles west.
Since ship B is sailing at a constant speed of 25 knots northward from noon to 6 PM, the distance traveled by ship B is 25 knots multiplied by 6 hours, which equals 150 nautical miles north.
Step 2: Use the Pythagorean theorem to find the distance between the ships at 6 PM.
The distance between the ships can be calculated using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
In this case, the distance between the ships is the hypotenuse of a right-angled triangle, with the westward distance traveled by ship A (a) as one side and the northward distance traveled by ship B (b) as the other side.
Using the Pythagorean theorem:
Distance between the ships = √(a^2 + b^2)
Plugging in the values:
Distance between the ships = √(150^2 + 150^2)
Distance between the ships = √(22500 + 22500)
Distance between the ships = √45000
Distance between the ships ≈ 212.13 nautical miles
Step 3: Calculate the rates of change for ship A and ship B separately.
The rate of change of distance for ship A is the rate of change of the westward distance traveled, which is a constant 25 knots.
The rate of change of distance for ship B is the rate of change of the northward distance traveled, which is also a constant 25 knots.
Step 4: Find the rate of change of the distance between the ships at 6 PM.
To find the rate of change of the distance between the ships, we need to find the derivative of the distance between the ships with respect to time (t), t being the time elapsed in hours from noon to 6 PM.
Differentiating the distance between the ships (d) with respect to time (t):
dd/dt = (√(a^2 + b^2))'
Using the chain rule:
dd/dt = (1/2) * (a^2 + b^2)^(-1/2) * (2a * a' + 2b * b'),
where a' is the rate of change of distance for ship A and b' is the rate of change of distance for ship B.
Plugging in the values:
dd/dt = (1/2) * (√(150^2 + 150^2))^(-1/2) * (2 * 150 * 25 + 2 * 150 * 25)
dd/dt = (1/2) * (212.13)^(-1/2) * (20,000 + 20,000)
dd/dt ≈ 282.84 knots
Therefore, the speed at which the distance between the ships is changing at 6 PM is approximately 282.84 knots.
To find the speed at which the distance between the ships is changing, we can use the concept of relative velocity.
Let's first determine the position of each ship at 6 PM. At noon, ship A was 30 nautical miles due west of ship B.
Since ship A is sailing west at 25 knots for 6 hours, its position at 6 PM would be 30 + (25 * 6) = 180 nautical miles west of its starting point.
Similarly, ship B is sailing north at 25 knots for 6 hours, so its position at 6 PM would be (25 * 6) = 150 nautical miles north of its starting point.
Now, we can determine the distance between the ships at 6 PM using the Pythagorean theorem, which states that the square of the hypotenuse (the distance between the ships) is equal to the sum of the squares of the other two sides (the distances traveled by each ship).
The distance between the ships squared = (180)^2 + (150)^2
Now, we need to find the rate at which this distance is changing. To do that, we can differentiate the equation with respect to time.
Differentiating the equation gives us:
2 * (distance between the ships) * (rate of change of distance) =
2 * (180) * (rate of change of 180) + 2 * (150) * (rate of change of 150)
Simplifying this equation, we have:
(rate of change of distance) = [(180) * (rate of change of 180) + (150) * (rate of change of 150)] / (distance between the ships)
To find the rate of change of 180 and 150, we can differentiate the positions of each ship with respect to time:
(rate of change of 180) = 25 knots (since ship A is moving west at a constant speed of 25 knots)
(rate of change of 150) = 25 knots (since ship B is moving north at a constant speed of 25 knots)
Now, we can substitute these values into our equation:
(rate of change of distance) = [(180) * (25) + (150) * (25)] / (distance between the ships)
Plugging in the values:
(rate of change of distance) = (4500 + 3750) / (distance between the ships)
Finally, we can find the distance between the ships using the Pythagorean theorem:
(distance between the ships) = sqrt[(180)^2 + (150)^2]
Substituting this value:
(rate of change of distance) = (4500 + 3750) / sqrt[(180)^2 + (150)^2]
Calculating this expression will give us the speed at which the distance between the ships is changing at 6 PM.