Ship A, 15 miles of 0 is moving west at 20 mile per hour, ship B, 60 miles south of 0 is moving north at 15 mile per hour. are they approaching or separating after 1 hr and at what rate? are they approaching or separating after 3 hrs? when are they nearest to one another? kindly illustrate only.
To determine if the ships are approaching or separating after a certain period of time, and at what rate, we need to calculate their distances from each other at different times.
Let's start by finding the position of each ship at the beginning, which will be at time t = 0.
Ship A is 15 miles to the west of point 0, so its initial position is (-15, 0).
Ship B is 60 miles to the south of point 0, so its initial position is (0, -60).
Now let's calculate their positions after 1 hour.
Since Ship A is moving west at a speed of 20 miles per hour, its position after 1 hour will be (-15 - 20*1, 0) = (-35, 0).
Similarly, as Ship B is moving north at a speed of 15 miles per hour, its position after 1 hour will be (0, -60 + 15*1) = (0, -45).
Now, we can calculate the distance between the two ships after 1 hour using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Distance = sqrt((0 - (-35))^2 + (-45 - 0)^2)
Distance = sqrt(35^2 + 45^2)
Distance = sqrt(1225 + 2025)
Distance = sqrt(3250)
Distance ≈ 57.0089 miles
So after 1 hour, the ships are approximately 57.0089 miles apart.
To determine if the ships are approaching or separating, we compare their distances at different times. If the distance between them is decreasing, they are approaching. If the distance is increasing, they are separating.
After 3 hours, we calculate their positions and find the distance between them again:
Ship A's position after 3 hours: (-15 - 20*3, 0) = (-75, 0)
Ship B's position after 3 hours: (0, -60 + 15*3) = (0, -15)
Distance after 3 hours:
Distance = sqrt((-75 - 0)^2 + (0 - (-15))^2)
Distance = sqrt(75^2 + 15^2)
Distance = sqrt(5625 + 225)
Distance = sqrt(5850)
Distance ≈ 76.4615 miles
Comparing the distances, we can see that the distance between the ships after 3 hours (76.4615 miles) is greater than the distance after 1 hour (57.0089 miles). Therefore, the ships are separating after 3 hours.
To determine when they are nearest to one another, we need to find the time at which the distance between them is minimized. This occurs when the ships are moving directly towards each other, causing the distance to decrease.
In this scenario, the x-coordinate of Ship A's position (westward distance) must be equal to the x-coordinate of Ship B's position (eastward distance), and the y-coordinate of Ship A's position (northward distance) must be equal to the y-coordinate of Ship B's position (southward distance).
Equating the x-coordinates:
-15 - 20t = 0 (where t is the time in hours)
t = -15 / -20
t = 0.75 hours
Equating the y-coordinates:
-60 + 15t = 0
15t = 60
t = 60 / 15
t = 4 hours
Therefore, the ships will be nearest to each other after 0.75 hours, or 45 minutes, when they are approximately 57.0089 miles apart.
I hope this explanation helps illustrate the positions, distances, and relationships between the two ships.