Problems

Solution: The position of the ship is \(\mathbf{s} = \binom{20t}{1}\). Suppose the interception is at \(T\), then the ship leaves at \(T-\frac1{12}\underbrace{\sqrt{400T^2+1}}_{\text{distance to intercept}}\). We wish to maximise this, ie \begin{align*} && \frac{\d}{\d T} \left ( T - \frac1{12}\sqrt{400T^2+1}\right) &= 1 - \frac{1}{12} \cdot \frac12 \cdot 400 \cdot 2T \cdot \left (400T^2+1 \right)^{-1/2} \\ &&&= 1 - \frac{100}3 T(400T^2+1)^{-1/2} \\ \Rightarrow && \frac{T}{\sqrt{400T^2+1}} &= \frac{3}{100} \\ \Rightarrow && \frac{T^2}{400T^2+1} &= \frac{9}{10000} \\ \Rightarrow && 10000T^2 &= 3600T^2+9 \\ \Rightarrow && 6400T^2 &= 9 \\ \Rightarrow && T &= \pm \frac{3}{80} \quad \text{(T > 0)} \end{align*} Therefore the distance is \(\sqrt{400 \frac{9}{6400} + 1} = \sqrt{\frac{9}{16}+1} = \frac{5}{4} = 1.25 \text{ km}\)