At 1:00 p.m. ship A is 25 km due north of ship B. If ship A is sailing west at a rate of 16km/h and ship B is sailing south at 20km/h What is the first time after 3:00 p.m. that the hands of the clock are together?
Well, it seems like the hands of the clock weren't exactly invited to this party between ship A and ship B. However, let's try to find out when they accidentally bump into each other!
Ship A is sailing west at a rate of 16 km/h, and ship B is sailing south at 20 km/h. So, ship A is moving towards the left (west) and ship B is moving downwards (south). Since we want to know the time when they meet, we need to figure out the distance they cover before this happens.
From 1:00 p.m. to 3:00 p.m., we have a time difference of 2 hours. This means ship A has been sailing west for 2 hours at a speed of 16 km/h. So, the distance covered by ship A is 32 km (2 hours multiplied by 16 km/h).
On the other hand, ship B has been sailing south for 2 hours at a speed of 20 km/h. Hence, the distance covered by ship B is 40 km (2 hours multiplied by 20 km/h).
Now, let's imagine their meeting point. Since ship A started 25 km north of ship B, we can say that ship A needs to cover an additional 25 km towards the south to meet ship B. Similarly, ship B needs to cover an additional 32 km towards the west to meet ship A.
So, ship A needs to cover 57 km in total (25 km towards the south + 32 km towards the west), and ship B needs to cover 72 km in total (40 km towards the west + 32 km towards the south).
To find out when they meet, we need to figure out which ship will reach this total distance first. Let's do some math:
Ship A's speed is 16 km/h. Therefore, the time it takes ship A to cover 57 km is 57 divided by 16, which is approximately 3.56 hours.
Ship B's speed is 20 km/h. Therefore, the time it takes ship B to cover 72 km is 72 divided by 20, which is approximately 3.6 hours.
Since the time should be after 3:00 p.m., we can round up ship A's time to 4 hours.
So, ship A and ship B will meet for the first time after 3:00 p.m. at approximately 4:00 p.m. Now, they can finally introduce the clock hands and have a proper gathering!
To determine the first time after 3:00 p.m. that the hands of the clock are together, we need to find the time when ship A and ship B are at the same location.
First, let's calculate the position of ship A and ship B at 3:00 p.m:
Since ship A is sailing west at a rate of 16 km/h, it has covered a distance of 16 km/h x 2 h = 32 km by 3:00 p.m.
Since ship B is sailing south at a rate of 20 km/h, it has covered a distance of 20 km/h x 2 h = 40 km by 3:00 p.m.
Now, we can form a right triangle to represent the positions of ship A, ship B, and the final position where their hands are together.
Let's consider the distance between ship A and ship B as the hypotenuse of the right triangle.
Using Pythagorean theorem, we can find the distance between ship A and ship B:
Distance between ship A and ship B = √(32 km)^2 + (40 km)^2
Distance between ship A and ship B = √(1024 km^2 + 1600 km^2)
Distance between ship A and ship B = √2624 km^2
Distance between ship A and ship B ≈ 51.22 km
Since ship A is 25 km due north of ship B, the remaining distance between them is 51.22 km - 25 km = 26.22 km.
Now, let's calculate the time required for ship A and ship B to cover this remaining distance:
Since ship A is sailing west at a rate of 16 km/h, it will take ship A 26.22 km / 16 km/h ≈ 1.64 hours to cover this distance.
Therefore, the first time after 3:00 p.m. that the hands of the clock are together is 3:00 p.m. + 1.64 hours ≈ 4:38 p.m.
To determine the first time after 3:00 p.m. that the hands of the clock are together, we need to calculate the relative distance and speed between the two ships in order to find out when they will meet.
Given that Ship A is sailing west at a rate of 16 km/h and Ship B is sailing south at 20 km/h, we can break down their movements into a coordinate system. Let's consider the position of Ship B as the origin (0, 0) on a graph, with the x-axis representing the west and east direction, and the y-axis representing the north and south direction.
At 1:00 p.m., Ship A is 25 km due north of Ship B. This means that Ship A's position on the graph is (0, 25). Since we are interested in finding when the hands of the clock are together after 3:00 p.m., we'll consider the time from 3:00 p.m. onwards.
Now, let's calculate the distance between the two ships using their relative speeds:
- From 1:00 p.m. to 3:00 p.m., which is a duration of 2 hours, Ship A travels west for a distance of 16 km/h * 2 h = 32 km.
- During the same duration, Ship B moves south for a distance of 20 km/h * 2 h = 40 km.
At 3:00 p.m., Ship A's position on the graph is (-32, 25) and Ship B's position is (0, -40).
To find the time when the hands of the clock are together, we need to determine when the x-coordinate of Ship A equals the x-coordinate of Ship B. This is the point where both ships meet.
The x-coordinate of Ship A can be given as -32 + (16 * t), where t is the time measured in hours starting from 3:00 p.m. The x-coordinate of Ship B remains zero.
Setting the two x-coordinates equal to each other, we get the equation:
-32 + (16 * t) = 0
Simplifying the equation by moving -32 to the other side:
16 * t = 32
Dividing both sides by 16:
t = 2
Therefore, the first time after 3:00 p.m. that the hands of the clock are together is 2 hours after 3:00 p.m., which is 5:00 p.m.