A cross country race course on flat ground has runners heading from point A for 3.5 km on a bearing of 047degT to point B. From point B they turn and run for 2.8 km on a bearing of 342degT to point C. They then run back to point A.

Determine the Distance from C back to A.

they run on a heading, not a bearing.

Starting at (0,0)
3.5 at 47° = (2.56,2.39)
2.8 at 342° = (-0.87,2.66)

Add them together to see that C is at (1.49,5.05)

CA = √(1.49^2+5.05^2) = 5.27

To determine the distance from point C back to point A, we need to visualize the race course and apply basic trigonometry.

1. Let's start by drawing a diagram of the race course.
Note: "degT" refers to degrees true, which is the direction measured clockwise from North.

B
/
/
/
/
A ---------------------- C

Point A is at the starting line, point B is 3.5 km away along a bearing of 047 degrees true, and point C is 2.8 km away from B along a bearing of 342 degrees true.

2. To find the distance from C back to A, we need to calculate the horizontal and vertical components of the distances traveled.

3. Horizontal component:
The horizontal distance traveled from point B to C is given by BC * cos(angle).
BC = 2.8 km (given)
Angle = 180 degrees - 342 degrees (since we are moving in the opposite direction)
= -162 degrees

BC_horiz = 2.8 km * cos(-162 degrees)

4. Vertical component:
The vertical distance traveled from point B to C is given by BC * sin(angle).
BC_vert = 2.8 km * sin(-162 degrees)

5. Now, we need to determine the displacement from C back to A.
The horizontal and vertical components calculated in steps 3 and 4 form a right-angled triangle with the displacement as the hypotenuse.
We can use the Pythagorean theorem to find the total displacement.

Displacement = √(BC_horiz^2 + BC_vert^2)

6. Finally, we can substitute the values we calculated into the equation to find the distance from C back to A.

Distance from C back to A = √(BC_horiz^2 + BC_vert^2)

Following these steps, you can calculate the distance from C back to A by substituting the values for BC_horiz and BC_vert into the equation and then solving it.