Answer:
A) Vector component for the cruise ship: (0, -22)
B) Vector component for the Gulf Stream: (4, 0)
C) Resultant vector: (4, -22)
D) Resultant velocity: Approximately 22.4 mph (rounded to the nearest tenth)
E) Resultant direction: Approximately -80.5 degrees (rounded to the nearest tenth)
Explanation:
To solve this problem, we'll consider the velocities of the cruise ship and the Gulf Stream as vectors and calculate their components and resultant vector. Then we'll find the magnitude (resultant velocity) and direction (resultant direction) of the resultant vector.
Given:
Cruise ship velocity (south): 22 mph
Gulf Stream velocity (east): 4 mph
A) Vector component for the cruise ship:
The cruise ship is traveling south, so its velocity vector is (0, -22).
B) Vector component for the Gulf Stream:
The Gulf Stream is flowing east, so its velocity vector is (4, 0).
C) Resultant vector:
To find the resultant vector, we'll add the two velocity vectors together:
Resultant vector = Cruise ship velocity + Gulf Stream velocity
Resultant vector = (0, -22) + (4, 0)
Resultant vector = (0 + 4, -22 + 0)
Resultant vector = (4, -22)
D) Resultant velocity:
The magnitude of the resultant vector gives us the resultant velocity. We can use the Pythagorean theorem to calculate it:
Resultant velocity = sqrt((x-component)^2 + (y-component)^2)
Resultant velocity = sqrt((4)^2 + (-22)^2)
Resultant velocity = sqrt(16 + 484)
Resultant velocity = sqrt(500)
Resultant velocity ≈ 22.4 mph (rounded to the nearest tenth)
E) Resultant direction:
The direction of the resultant vector can be found using trigonometry. We'll use the inverse tangent function (arctan) to find the angle between the resultant vector and the positive x-axis.
Resultant direction = arctan(y-component / x-component)
Resultant direction = arctan(-22 / 4)
Resultant direction ≈ -1.405 radians or -80.5 degrees (rounded to the nearest tenth)
Therefore, the answers are:
A) Vector component for the cruise ship: (0, -22)
B) Vector component for the Gulf Stream: (4, 0)
C) Resultant vector: (4, -22)
D) Resultant velocity: Approximately 22.4 mph (rounded to the nearest tenth)
E) Resultant direction: Approximately -80.5 degrees (rounded to the nearest tenth)