Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks 28.0 in a direction 60.0 west of north. Jane walks 16.0 in a direction 30.0 south of west. They then stop and turn to face each other.
To find the direction in which Ricardo and Jane are facing each other, we can use vector addition to determine their relative positions.
First, let's convert the given directions into their respective components.
Ricardo's direction:
- Magnitude: 28.0
- Angle: 60.0° west of north
To convert this to Cartesian coordinates, we need to break down the magnitude into its components using trigonometry. The horizontal component (north-south direction) can be found using the sine function, and the vertical component (west-east direction) can be found using the cosine function.
Horizontal component of Ricardo's direction:
- Magnitude: 28.0 * sin(60.0°) = 24.2 (rounded to one decimal place)
- Direction: North
Vertical component of Ricardo's direction:
- Magnitude: 28.0 * cos(60.0°) = 14.0 (rounded to one decimal place)
- Direction: East
Jane's direction:
- Magnitude: 16.0
- Angle: 30.0° south of west
Again, let's convert this to Cartesian coordinates.
Horizontal component of Jane's direction:
- Magnitude: 16.0 * cos(30.0°) = 13.9 (rounded to one decimal place)
- Direction: East
Vertical component of Jane's direction:
- Magnitude: 16.0 * sin(30.0°) = 8.0 (rounded to one decimal place)
- Direction: South
Now, let's determine their positions.
Ricardo's position:
- Horizontal component: 14.0 (east)
- Vertical component: 24.2 (north)
Jane's position:
- Horizontal component: 13.9 (east)
- Vertical component: -8.0 (south)
To find the relative position of Ricardo with respect to Jane, we can subtract Jane's position vector from Ricardo's position vector.
Relative position (vector addition):
- Horizontal component: 14.0 - 13.9 = 0.1 (east)
- Vertical component: 24.2 - (-8.0) = 32.2 (north)
Now, we can find the direction of the relative position vector using the inverse tangent (arctan) function.
Relative position direction:
- Angle: arctan(0.1 / 32.2) = 1.77°
Therefore, Ricardo and Jane are facing each other in a direction approximately 1.77° north of east.
There should be units on these numbers, so for the sake of the question I will use meters. So this is a problem of a right triangle, 60 degrees plus 30 degrees is 90 degrees, the distances they travel are two sides of the triangle, and the distance between them when they stop and face each other is the third side so let's use the pythagorian theorem:
Side 1(a)= 28.0 m
Side 2(b)= 16.0 m
Hypotenuse = c
a^2 + b^2 = c^2
(28.0)^2 + (16.0)^2 = c^2
784 + 256 = c^2
1040 = c^2
1040^(1/2) = c
c = 32.25 m
When Ricardo and Jane stopped to face each other they were 32.25 meters (or whatever unit you were given) away from each other