The gap between the rails

Steel rails $13\ \text{m}$ long are laid at $15\ ^\circ\text{C}$ with a small gap between them. Through the year the rails range from $-10\ ^\circ\text{C}$ to $60\ ^\circ\text{C}$. You will make the program find how the gap changes, and the smallest gap at laying that never closes.

The idea. Each rail grows by $\alpha L(T - T_{\text{laid}})$, and each gap shrinks by the same amount. With $\alpha = 1.2 \times 10^{-5}\ \text{K}^{-1}$.

Your tasks.

1. Work out how much a rail has grown, growth, at every temperature in T (negative when colder than at laying).
2. Work out the gap at every temperature, starting from gap0.
3. Find the smallest gap at laying, gap_needed, that is still open at $60\ ^\circ\text{C}$.