A ship started sailing 42.58 degrees west of south at the rate of 15 kph. after 2 hours, ship B started at the same port going 46.33 degrees west of north at the rate of 7 kph. after how many hours will the second ship be exactly north of ship A?
you are just interested in the east-west distances traveled. When they are the same, B is north of A. So, t hours after A set off, you want
(15 sin42.58°)t = (7 sin46.33°)(t-2)
This has a solution of t = -2
huh? I suspect a typo. A is going twice as fast, for two hours longer, and at almost the same E-W speed. No way could B catch up to it.
To determine when the second ship will be exactly north of ship A, we need to calculate the time it takes for the second ship to cover the north-south distance between the two ships.
Let's break down the problem into components:
1. Ship A is sailing 42.58° west of south at a speed of 15 kph. This means that it is moving in a direction that is 42.58° clockwise (west) from the south direction.
2. Ship B is sailing 46.33° west of north at a speed of 7 kph. This means it is moving in a direction that is 46.33° counterclockwise (west) from the north direction.
Now, we need to find the north-south distance between the two ships. Since both ships start at the same port, this distance will remain constant.
We can use trigonometry to find the north-south component for both ships:
For Ship A:
North-South Component (A) = Speed (A) * sin(Direction (A)) = 15 kph * sin(180° - 42.58°) = 15 kph * sin(137.42°)
For Ship B:
North-South Component (B) = Speed (B) * sin(Direction (B)) = 7 kph * sin(360° - 46.33°) = 7 kph * sin(313.67°)
To calculate when Ship B will exactly be north of Ship A, we need to find the time it takes for the north-south components to be equal, i.e., North-South Component (A) = North-South Component (B).
15 kph * sin(137.42°) = 7 kph * sin(313.67°)
To solve this equation, we need to find the time taken by Ship B.
Time taken by Ship B = Speed (B) / North-South Component (B) = 7 kph / (7 kph * sin(313.67°) / sin(137.42°))
Simplifying the equation:
Time taken by Ship B = 1 / (sin(313.67°) / sin(137.42°))
Using a scientific calculator or math software to evaluate this equation, we find that the time taken by Ship B is approximately 5.04 hours. Therefore, after around 5.04 hours, Ship B will be exactly north of Ship A.