At 12 noon ship A is 60 miles west of point P steaming east at 15 knots and ship B is 36 miles south of P steaming north at 10 knots. If the ships continue their courses and speed. how is the distance between them changes at 2pm? At what time the ships are closest?
The answers are 17.94 mph and 3.53pm, but i don't know how to get that. I convey knots to mph and my answers is always around 20.61 mph. Please help.
Why are you looking for speeds? The question is about distance.
Let's assume that the miles are nautical miles, so no conversions are necessary.
At time t hours, the distance z is found using
z^2 = (60-15t)^2 + (36-10t)^2
= 325t^2 - 2520t + 4896
2z dz/dt = 650t - 2520
The ships are closest when
dz/dt = 0 at t=3.877 = 3:52 pm
at 2:00, (t=2), z = 30.26
So, 60.52 dz/dt = 1300-2520
dz/dt = -20.15
so the ships are approaching each other at 20.15 knots
Actually, I think that the actual conversion ratio of knots to mi/hr is irrelevant, since the same scale factor will be used throughout.
To solve this problem, we can use the concept of relative velocity. The relative velocity is the velocity of one object with respect to the other.
1. Converting knots to mph:
Since the question provides the speeds of the ships in knots, we need to convert them to miles per hour (mph).
1 knot is equal to 1.15077945 mph.
So, ship A is traveling at 15 knots * 1.15077945 mph ≈ 17.26 mph,
and ship B is traveling at 10 knots * 1.15077945 mph ≈ 11.51 mph.
2. Finding the relative velocity:
To calculate the distance between the ships at any given time, we need to find the relative velocity between them.
Ship A is moving east, while Ship B is moving north.
Using the Pythagorean theorem, we can find the relative velocity:
Relative Velocity = √((Velocity of Ship A)^2 + (Velocity of Ship B)^2)
Relative Velocity = √((17.26 mph)^2 + (11.51 mph)^2)
Relative Velocity ≈ √(298.1476 + 132.8601)
Relative Velocity ≈ √431.0077
Relative Velocity ≈ 20.77 mph (rounded to two decimal places)
Therefore, the rate at which the distance between the ships changes is approximately 20.77 mph.
3. Finding the time when the ships are closest:
We know that the ships are currently 60 miles west and 36 miles south of the point P.
The time it takes for the ships to meet can be calculated by dividing the initial distance between them by their relative velocity.
Initial distance between the ships = √((60 miles)^2 + (36 miles)^2)
Initial distance between the ships ≈ √(3600 + 1296)
Initial distance between the ships ≈ √4896
Initial distance between the ships ≈ 69.96 miles (rounded to two decimal places)
Time taken to meet = Initial distance / Relative Velocity
Time taken to meet ≈ 69.96 miles / 20.77 mph
Time taken to meet ≈ 3.36 hours (rounded to two decimal places)
To find the time when the ships are closest, we need to add the time taken to meet to the initial time when the ships started (12 noon), which gives us:
Closest time = Initial time + Time taken to meet
Closest time ≈ 12 noon + 3.36 hours
Closest time ≈ 3.36 pm (rounded to two decimal places)
Therefore, the time when the ships are closest is approximately 3.36 pm.
To find the distance between the two ships at 2 pm, we can break down the problem into two components: the horizontal distance and the vertical distance.
Let's first consider the horizontal distance. Ship A is steaming east at 15 knots for 2 hours (from 12 pm to 2 pm), so it covers a horizontal distance of 15 knots/hour * 2 hours = 30 miles to the east of point P. Ship B, on the other hand, is stationary in the east-west direction, so its horizontal position does not change. Therefore, at 2 pm, the horizontal distance between the two ships is 60 miles (initial distance between A and P) + 30 miles (distance covered by A) = 90 miles.
Now let's consider the vertical distance. Ship B is steaming north at 10 knots for 2 hours, so it covers a vertical distance of 10 knots/hour * 2 hours = 20 miles to the north of point P. Ship A, being stationary in the north-south direction, does not change its vertical position. Therefore, at 2 pm, the vertical distance between the two ships is 36 miles (initial distance between B and P) + 20 miles (distance covered by B) = 56 miles.
To find the total distance between the two ships, we can use the Pythagorean theorem. The distance is given by the square root of the sum of the squares of the horizontal and vertical distances. In this case, the distance is √(90^2 + 56^2) ≈ 107.352 miles.
To convert knots to mph, we can use the conversion factor: 1 knot = 1.15078 mph. Therefore, 15 knots is approximately 17.26 mph, and 10 knots is approximately 11.51 mph.
To find the rate at which the distance between the ships changes, we can use the concept of relative velocity. The two ships are moving towards each other, so the rate at which the distance decreases is the sum of their speeds: 17.26 mph + 11.51 mph = 28.77 mph.
To find the time when the ships are closest, we need to find the point at which the rate of change of distance is zero. We can calculate the time using the formula: time = initial distance / relative speed. In this case, the initial distance between the ships is 90 miles, and the relative speed is 28.77 mph. Therefore, the time when the ships are closest is 90 miles / 28.77 mph ≈ 3.126 hours.
To convert this to minutes, we multiply by 60: 3.126 hours * 60 minutes/hour ≈ 187.56 minutes. Rounding this to the nearest whole number, we get that the ships are closest at approximately 188 minutes, which is 3 hours and 8 minutes.
So, to summarize, the distance between the two ships at 2 pm is approximately 107.352 miles. The rate at which the distance changes is approximately 28.77 mph, and the ships are closest at around 3:08 pm.