A ship sails 300 km due west and then 100 km due south. At the end of this journey, how far is the ship from its starting position?

Step 1: The journey forms a right triangle with legs of 300 km (west) and 100 km (south).
The straight-line distance $d$ from start to end is the hypotenuse:
$d = \sqrt{(300~km)^2 + (100~km)^2}$

Step 2: Square the distances.
$(300)^2 = 90{,}000$
$(100)^2 = 10{,}000$

Step 3: Add the squares.
$90{,}000 + 10{,}000 = 100{,}000$

Step 4: Take the square root.
$d = \sqrt{100{,}000} = \sqrt{10{,}000 \times 10} = 100\sqrt{10}~\text{km}$

Final answer: $\boxed{100\sqrt{10}~km}$