A hiker travels at 4 miles per hour at a heading of S 35° E from a ranger station. After 3 hours how far south and how far east is the hiker from the station?

The magnitude of southern displacement of the hiker is 6.88 miles.

The magnitude of eastern displacement of the hiker is 9.83 miles.

The given parameters;

The resultant displacement of the hiker is calculated as follows;

D = vt

D = 4 mph x 3 h

D = 12 miles

The magnitude of southern displacement of the hiker is calculated as follows;

The magnitude of eastern displacement of the hiker is calculated as follows;

Learn more about resultant displacement here: link

First, let's define our coordinates.

Let's call East as the positive x-axis and North as the positive y-axis.

Now the data that we have is:

Speed  = 4 mph

Time = 3 hours.

Then the total distance traveled is:

D = 4mph*3h = 12 miles.

We also know that the angle is S 35° E.

(from east, 35° to south)

So we can write this angle as -35°.

Then if we want to know the component in South and East, we can first think this in polar coordinates

(R, θ)

where R = 12 miles, and θ = -35°

Now with this, we have that:

X = R*cos(θ) = the displacement to the east.

Y = R*sin(θ) = the displacement to the north (then -Y is the displacement to the south).

Then we have:

X = 12mi*cos(-35°) = 9.83mi

Y = 12mi*sin(-35°) = - 6.89 mi.

Then:

The displacement to the East is X = 9.83 miles

The displacement to the South is -Y = 6.89 miles.