A lighthouse stands at the origin of a map, with distances in kilometres…

Step 1. The condition. The distance is smallest when $\vec r$ is at right angles to $\vec v$. So $\vec r\cdot\vec v = 0$.

Step 2. Solve for $t$. $\vec r\cdot\vec v = (-25 + 3t)(3) + (4t)(4) = -75 + 25t = 0$, so $t = 3$ hours.

Step 3. The position. $\vec r = (-25 + 9, 12) = (-16, 12)$.

Step 4. The distance. $\lvert\vec r\rvert = \sqrt{256 + 144} = 20$ kilometres.

Step 5. The speed. $\lvert\vec v\rvert = \sqrt{9 + 16} = 5$ kilometres per hour.