A woman runs 12 km towards her North, then 6 km towards her South and then 8 km towards her East. In which direction is she from her starting point?

Correct Answer: An angle less than 45° North of East.

Step-by-step Explanation:

Step 1: Track the movement along the North-South axis

The woman first runs 12 km towards the North.

Then, she turns back and runs 6 km towards the South.

Her net displacement along the North-South axis is:

$\text{Net North Displacement}=12\text{km}-6\text{km}=6\text{km North}$

Step 2: Track the movement along the East-West axis

Next, she runs 8 km towards the East.

So, her displacement along the East-West axis is:

$\text{Net East Displacement}=8\text{km East}$

Step 3: Determine the direction relative to the starting point

Since her net displacement is 6 km to the North and 8 km to the East, she is in the North-East quadrant relative to her starting point.

Let $\theta$ be the angle of her position North of East. The angle $\theta$ can be calculated using the tangent function:

$\tan \left(\theta \right)=\frac{\text{Opposite (North displacement)}}{\text{Adjacent (East displacement)}}$

$\tan \left(\theta \right)=\frac{6}{8}=0.75$

Step 4: Compare with 45°

We know that $\tan (45°)=1$.

Since $0.75<1$, the angle $\theta$ must be less than 45°.

Conclusion:

The woman is at an angle less than 45° North of East from her starting point.