Problem: $($Show all steps in your solution$)$

Consider the following scenario: A dog is initially positioned at $(0,30)m),$ while a fox is located at $(40,0)m.$ The fox moves along the $x-$axis at a constant speed of $v=12\frac{m}{s},$ while the dog continually adjusts its trajectory to aim directly at the fox at every instant. The dog's speed is $u=15\frac{m}{s}.$

Determine the time it takes for the dog to catch the fox.