A man travels 12km towards west then 3 km towards south and then 8 km towards east. How far is he from the Start?

D = 5km[37o] S. of W.

D = -12 - 3i + 8 = -4 - 3i = 5km[37o].

To find how far the man is from the start, we can use the concept of distance and direction.

First, let's plot the movements on a coordinate plane. Assume the starting point as the origin (0,0).

- The man travels 12 km towards the west. Moving to the west means moving along the x-axis in the negative direction. So, his position now is (-12,0).

- Next, he travels 3 km towards the south. Moving south means moving along the y-axis in the negative direction. So, his position now is (-12, -3).

- Finally, he travels 8 km towards the east. Moving east means moving along the x-axis in the positive direction. So, his position now is (-12+8, -3) = (-4, -3).

To find the distance from the start (origin) to the man's current position, we can use the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case:
x1 = 0 (starting x-coordinate)
y1 = 0 (starting y-coordinate)
x2 = -4 (final x-coordinate)
y2 = -3 (final y-coordinate)

Distance = sqrt(((-4) - 0)^2 + ((-3) - 0)^2)
= sqrt((-4)^2 + (-3)^2)
= sqrt(16 + 9)
= sqrt(25)
= 5 km

Therefore, the man is 5 km away from the start.

his final location is 4 east, 3 south

so the distance is 5km