Two ships sail from the same port. The first ship leaves port at 1:00a, and travels eastward at a rate of 15 knots. the second ship leave port at 2:00am and travels northward at a rate of 10 knots. Determine the rate at which the ships are separating at 3:00am
To determine the rate at which the ships are separating at 3:00am, we can use the concept of relative velocity.
Step 1: Find the distance traveled by each ship from the time it left port until 3:00am.
The first ship left at 1:00am and traveled for 2 hours (until 3:00am). Since the ship is traveling eastward at a rate of 15 knots, the distance it traveled can be calculated as:
Distance of the first ship = 15 knots/hour * 2 hours = 30 knots
The second ship left at 2:00am and traveled for 1 hour (until 3:00am). Since the ship is traveling northward at a rate of 10 knots, the distance it traveled can be calculated as:
Distance of the second ship = 10 knots/hour * 1 hour = 10 knots
Step 2: Determine the separation distance between the two ships at 3:00am.
Since the ships are moving at right angles to each other (eastward and northward), we can use the Pythagorean theorem to find the separation distance.
Separation distance = √(Distance of the first ship)^2 + (Distance of the second ship)^2
Separation distance = √(30 knots)^2 + (10 knots)^2
Separation distance = √(900 knots^2 + 100 knots^2)
Separation distance = √(1000 knots^2)
Separation distance = 31.62 knots (approximately)
Step 3: Calculate the rate at which the ships are separating.
Since the separation distance is the hypotenuse of a right triangle formed by the distances traveled by each ship, we need to find the derivative of the separation distance with respect to time. This will give us the rate at which the separation distance is changing.
To do this, we can differentiate the separation distance equation with respect to time:
d(separation distance)/dt = (d/dt)√((Distance of the first ship)^2 + (Distance of the second ship)^2)
To simplify this, we can apply the chain rule:
d(separation distance)/dt = (1/2)*(Distance of the first ship / √(Distance of the first ship)^2 + (Distance of the second ship)^2) * (d/dt)(Distance of the first ship)^2 + (Distance of the second ship)^2)
Simplifying further, we get:
d(separation distance)/dt = (1/2)*(Distance of the first ship) * (d/dt)(Distance of the first ship) / √((Distance of the first ship)^2 + (Distance of the second ship)^2)
Since the rate at which the first ship is traveling (15 knots) is constant, the derivative (d/dt) of the distance traveled by the first ship is zero.
Therefore, the rate at which the ships are separating at 3:00am is:
d(separation distance)/dt = (1/2)*(Distance of the first ship) * 0 / √((Distance of the first ship)^2 + (Distance of the second ship)^2)
d(separation distance)/dt = 0
Hence, the ships are not separating at 3:00am.
To determine the rate at which the ships are separating at 3:00 am, we first need to find their positions at that time.
Let's consider the first ship's position. It left the port at 1:00 am and travels eastward at a rate of 15 knots. Since it has been two hours since it left the port, the first ship will have traveled a distance of 15 knots/hour x 2 hours = 30 nautical miles east.
Now let's consider the second ship's position. It left the port at 2:00 am and travels northward at a rate of 10 knots. Since it has been one hour since it left the port, the second ship will have traveled a distance of 10 knots/hour x 1hour = 10 nautical miles north.
Therefore, at 3:00 am, the first ship will be 30 nautical miles east of the port, and the second ship will be 10 nautical miles north of the port.
To determine the rate at which the ships are separating at 3:00 am, we can use the Pythagorean theorem. The ships' separation distance is the hypotenuse of a right triangle formed by their horizontal and vertical distances from the port.
Using the Pythagorean theorem:
Separation distance^2 = (30 nautical miles)^2 + (10 nautical miles)^2
Separation distance^2 = 900 nautical miles^2 + 100 nautical miles^2
Separation distance^2 = 1000 nautical miles^2
Taking the square root of both sides:
Separation distance = sqrt(1000 nautical miles^2)
Separation distance ≈ 31.6 nautical miles
Therefore, at 3:00 am, the ships are separating at a rate of approximately 31.6 nautical miles.
the distance d between the ships is
d^2 = x^2 + y^2
2d dd/dt = 2x dx/dt + 2y dy/dt
at 3:00 am, x=30 and y=10, d=10√10
20√10 dd/dt = 2(30)(15) + 2(10)(10)
dd/dt = 1100/10√10 = 110/√10 = 11√10