Question

Ships a and b leave port together. For the next two hours, ship a travels at 20 mph in a direction 34° west of north while ship b travels 20° east of north at 25 mph. What is the distance between the two ships after two hours?

Answer 1

To find the distance between two ships after two hours, we calculate the respective distances they travel, determine their vector components, use vector subtraction, and apply the Pythagorean theorem to find the magnitude of the resultant displacement vector.

Step-by-step explanation:

We are asked to find the distance between two ships after two hours when they travel in different directions at different speeds. The movement of each ship can be represented as a vector, with the speed of the ship representing the magnitude of the vector and the direction representing the vector's direction. Ship A travels at 20 mph in a direction 34° west of north, while ship B travels at 25 mph 20° east of north.

To solve this, we will calculate the distance each ship travels and then use vector subtraction to determine the distance between them after two hours. Ship A travels 40 miles (20 mph × 2 hours), and Ship B travels 50 miles (25 mph × 2 hours).

Ship A's vector components can be calculated as:

• Ax = 40 cos(34°)
• Ay = 40 sin(34°)

Ship B's vector components can be calculated as:

• Bx = 50 cos(20°)
• By = 50 sin(20°)

We then find the distance between the ships by subtracting the components of Ship A from Ship B:

• Dx = Bx - Ax
• Dy = By - Ay

Finally, the

distance between the two ships

is the magnitude of the resultant vector, which we find using the Pythagorean theorem:

• Distance = √(Dx² + Dy²)