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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 10.0 in a direction 30.0 south of west. They then stop and turn to face each other

What is the distance between them? In what direction should Ricardo walk to go directly toward Jane?

Verified Answer :

MissPhiladelphia

For the answer to the question above,
Ricardo goes a distance (magnitude) of 27, in a direction of 60 degrees W of N 
Jane goes a magnitude of 16 in a direction 30 degrees S of W 

How I would solve this is to imagine that the started at (0,0) 
And their walking represents vectors. 

Ricardo: 
X-coordinate = -27sin60 = 27sqrt(3)/2 = 23.383 
Y-coordinate = 27cos60 = 27/2 = 13.5 
So, after he walks, he is at point (-23.383, 13.5) 

Jane: 
X-coordinate = -16cos(30) = 16sqrt(3)/2 = 13.856 
Y-coordinate = -16sin(30) = 16/2 = 8 
So, after she walks, she is at point (-13.856, -8) 

So, you have 2 points. 
Use the distance formula to find their distance apart 
Sqrt((-23.383+13.856)^2+(13.5+8)^2) = 23.516m 

To find the direction, simply find the slope of the two points, and take the arc-tangent. 
The slope = -9.527/21.5 = -0.443 
Take the tan^-1 of this, which is -23.899 degrees. 
This basically translates to, Ricardo must walk 23.899 degrees E of S 

They will be 23.518 m apart 
Ricardo must walk 23.899 degrees East of South to get to Jane

Answer:

Part a)

\(d = 24.5 m\)

Part b)

\(\theta = 39.2\) degree East of South

Explanation:

Let they both are at origin initially

so here we will have final coordinates of both of them is given as

Ricardo walk 28 m in direction of 60 degree West of North

\(x_1 = -28 sin60\)

\(x_1 = -24.2 m\)

\(y_1 = 28 cos60\)

\(y_1 = 14 m\)

Jane walks 10 m in direction 30 degree South of West

\(x_2 = -10 cos30\)

\(x_2 = -8.66 m\)

\(y_2 = -10 sin30\)

\(y_2 = -5 m\)

Part a)

distance between them is given as

\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

\(d = \sqrt{(-24.2 + 8.66)^2 + (14 + 5)^2}\)

\(d = 24.5 m\)

Part b)

direction of motion of Ricardo is given as

\(tan\theta = \frac{x_2 - x_1}{y_2 - y_1}\)

\(tan\theta = \frac{24.2 - 8.66}{14 + 5}\)

\(\theta = 39.2\) degree East of South

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