Fred got into his motorboat and motored for 12.5 km at a bearing of 30 degrees north of east. He then abruptly made a 150 degree right turn and motored for another 13.5 km to his destination. How far was his destination from his beginning point?

Question

Fred got into his motorboat and motored for 12.5 km at a bearing of 30 degrees north of east. He then abruptly made a 150 degree right turn and motored for another 13.5 km to his destination. How far was his destination from his beginning point?

My teacher is horrible at teaching and this has got something to do with vectors, but I don't know what vectors are.

Answer 1

To solve this problem, we can use vector addition to determine the distance between Fred's starting point and his destination. Vectors represent both direction and magnitude of a physical quantity, such as displacement or force, and can be represented using arrows.

First, let's break down Fred's movement into two separate vectors:

1. Fred moves 12.5 km at a bearing of 30 degrees north of east. This can be represented as a vector of 12.5 km at a 30-degree angle to the positive x-axis (east). We'll call this vector A.

2. After the abrupt right turn, Fred continues for 13.5 km. Since he turned 150 degrees to the right, his final bearing will be 30 degrees + 150 degrees = 180 degrees, pointing due west. This can be represented as a vector of 13.5 km at a 180-degree angle to the positive x-axis (west). We'll call this vector B.

Now, we can find the resultant vector by adding vectors A and B together. To do this, we'll separate them into their components, both in the x-direction (east-west) and the y-direction (north-south).

Vector A:
Magnitude: 12.5 km
Angle: 30 degrees north of east
Components:
Ax = 12.5 km * cos(30°)
Ay = 12.5 km * sin(30°)

Vector B:
Magnitude: 13.5 km
Angle: 180 degrees (due west)
Components:
Bx = 13.5 km * cos(180°)
By = 13.5 km * sin(180°)

Now let's calculate the components:

Ax = 12.5 km * cos(30°) = 12.5 km * √3/2 ≈ 10.825 km
Ay = 12.5 km * sin(30°) = 12.5 km * 1/2 = 6.25 km

Bx = 13.5 km * cos(180°) = -13.5 km
By = 13.5 km * sin(180°) = 0 km

To find the resultant vector, we can add the x-components together and the y-components together:

Rx = Ax + Bx
Ry = Ay + By

Rx = 10.825 km + (-13.5 km) ≈ -2.675 km
Ry = 6.25 km + 0 km = 6.25 km

Now, we can use the Pythagorean theorem to find the magnitude (distance) of the resultant vector:

R = √(Rx^2 + Ry^2)
R = √((-2.675 km)^2 + (6.25 km)^2)
R ≈ √(7.172 km^2 + 39.0625 km^2)
R ≈ √(46.234 km^2)
R ≈ 6.8 km

Therefore, Fred's destination is approximately 6.8 km from his starting point.