A girl starts at A and walks 2 km south to B. she then walks 3 km west to C. find the distance and bearing of C from A.
We can use the Pythagorean theorem and trigonometry to solve this problem.
First, let's draw a diagram of the situation:
```
A ---> B
|
V
C
```
From this diagram, we can see that the distance from A to C is the hypotenuse of a right triangle with legs 2 km (from A to B) and 3 km (from B to C). We can use the Pythagorean theorem to find the length of this hypotenuse:
```
c^2 = a^2 + b^2
c^2 = 2^2 + 3^2
c^2 = 4 + 9
c^2 = 13
c = sqrt(13) km
```
So the distance from A to C is approximately 3.61 km.
To find the bearing of C from A, we can use trigonometry. We know that the angle opposite the leg of length 3 km is a right angle (because we are assuming that B is directly south of A). So we can use the tangent function to find the angle θ between the line AC and the northern direction:
```
tan(θ) = opposite / adjacent
tan(θ) = 2 / 3
θ = arctan(2/3)
θ ≈ 33.7°
```
So the bearing of C from A is approximately N33.7°W.
To find the distance and bearing of point C from point A, we can use the Pythagorean theorem and trigonometric functions.
Step 1: Draw a diagram to visualize the situation. Point A is the starting point, point B is 2 km south of A, and point C is 3 km west of B.
Step 2: Calculate the distance from A to C using the Pythagorean theorem.
AC^2 = AB^2 + BC^2
AC^2 = (2 km)^2 + (3 km)^2
AC^2 = 4 km^2 + 9 km^2
AC^2 = 13 km^2
Taking the square root of both sides, we find:
AC = sqrt(13) km
Therefore, the distance from A to C is approximately 3.61 km.
Step 3: Determine the bearing of C from A.
To find the bearing, we can use the inverse tangent function.
tan(theta) = opposite/adjacent
tan(theta) = 3 km / 2 km
theta = arctan(3/2)
Using a calculator, we find:
theta ≈ 56.31 degrees
Therefore, the bearing of C from A is approximately 56.31 degrees.