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have look a at diagram above

=> $O$ is the starting point => $0<\alpha<\beta<\pi/2$ => $OL=OB=d$

i worked the distance $LB$ to be $2d\sin((\beta-\alpha)/2)$ but, i am struggling to understand this 'The distance between $L$ and $B$ when $L$ has walked a distance $d$ must be less than $kd$, where $k$ is some constant. Find $k$.

How do i find $k$ ?

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    If I am understanding the problem correctly, you are trying to find the maximum distance between L and B? – 2017-01-03
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    Yes, and for what value of 'k' will the distance LB will always be less than kd. d being the distance walked by L and B – 2017-01-04

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Why not $k\ge2\sin\left(\frac{\beta-\alpha}{2}\right)$? Or if you want something independent of $\alpha$ and $\beta$ use $\sin(\cdot) \le 1$ and take $k\ge 2$