A hiker travels N 33 degrees W for 15km, then turns 90 degrees and walks 12km in the direction S 57 degrees W At that time how far is the hiker from his starting point? What is his bearing from his starting point?

As usual, draw a diagram.

the distance is easy, since we have a right triangle

d^2 = 15^2 + 12^2 = 369
d = 19.2

The bearing is N (33+x) W where

tan x = 12/15
x = 37°

bearing is N 70° W

Alternatively, and with much more work,

(0,0) + (15 cos 123°,15 sin 123°) = (-8.17,12.58)
(-8.17,12.58) + (12 cos 213°, 12 sin213°) = (-8.17,12.58) + (-10.06,-6.54) = (-18.23,6.04)

d^2 = 369
tan x = -18.23/6.04 = -3.02
x = 180-20 = 160° = N 70° W

oops

tan x = 6.04/-18.23

To solve this problem, we can break it down into smaller steps.

Step 1: Determine the coordinates of the hiker after the first leg of the journey.
- The hiker travels N 33 degrees W for 15km, which means they move in a direction that is 33 degrees west of due north.
- To calculate the change in north and west coordinates, we use trigonometry. The north component is given by: 15 * sin(33) = 7.9 km (rounded to one decimal place). The west component is given by: 15 * cos(33) = 12.6 km (rounded to one decimal place).
- Since the hiker starts at the origin, their coordinates after the first leg are (12.6, 7.9).

Step 2: Determine the coordinates of the hiker after the second leg of the journey.
- The hiker then turns 90 degrees and walks 12km in the direction S 57 degrees W.
- This direction is 57 degrees west of due south.
- Again, we use trigonometry to calculate the change in south and west coordinates. The south component is given by: 12 * sin(57) = 9.2 km (rounded to one decimal place). The west component is given by: 12 * cos(57) = -6.3 km (rounded to one decimal place).
- Adding these components to the previously determined coordinates, the new coordinates of the hiker are (12.6 - 6.3, 7.9 - 9.2), which simplifies to (6.3, -1.3).

Step 3: Calculate the distance to the starting point.
- Using the distance formula, the distance to the starting point can be found as follows: sqrt((6.3)^2 + (-1.3)^2) = sqrt(39.94) = 6.3 km (rounded to one decimal place).
- Therefore, the hiker is approximately 6.3 km away from the starting point.

Step 4: Calculate the bearing from the starting point.
- To find the bearing, we use the inverse tangent function. The bearing is given by: atan(-1.3/6.3) = atan(-0.206) = -11.9 degrees (rounded to one decimal place).
- Since the bearing is negative, we add 360 degrees to it to obtain the final bearing: -11.9 + 360 = 348.1 degrees (rounded to one decimal place).
- Therefore, the hiker's bearing from the starting point is approximately 348.1 degrees.

To find out how far the hiker is from his starting point and his bearing, we can use basic trigonometry and vector addition.

First, let's break down the hiker's movements:

1. The hiker travels N 33 degrees W for 15km.
- This means the hiker moved 15km in the direction 33 degrees west of north.

2. The hiker then turns 90 degrees and walks 12km in the direction S 57 degrees W.
- This means the hiker moved 12km in the direction 57 degrees west of south.

Now, let's calculate the horizontal and vertical components of the hiker's movements.

For the first movement:
- Horizontal component: 15km * cos(33 degrees) = 12.55km
- Vertical component: 15km * sin(33 degrees) = 7.88km

For the second movement:
- Horizontal component: 12km * cos(57 degrees) = 6.67km
- Vertical component: -12km * sin(57 degrees) = -9.84km (negative because it is in the opposite direction)

Next, we add up the horizontal and vertical components to find the net displacement of the hiker from his starting point:

Horizontal displacement = 12.55km + 6.67km = 19.22km
Vertical displacement = 7.88km - 9.84km = -1.96km

We can use the Pythagorean theorem to find the total distance from the starting point:

Distance = √(Horizontal displacement^2 + Vertical displacement^2)
Distance = √(19.22km^2 + (-1.96km)^2)
Distance = √(370.42km + 3.84km)
Distance = √374.26km
Distance ≈ 19.35km

So, the hiker is approximately 19.35km away from his starting point.

To find the bearing from the starting point, we can use inverse trigonometry. The bearing is the angle between the direction of the displacement and the north direction.

Bearing = arctan(Vertical displacement / Horizontal displacement)
Bearing = arctan(-1.96km / 19.22km)
Bearing ≈ -5.81 degrees

Since the angle is negative, we need to add 360 degrees to get the positive bearing:

Bearing ≈ 360 degrees - 5.81 degrees
Bearing ≈ 354.19 degrees

So, the hiker's bearing from his starting point is approximately 354.19 degrees.