2007-I-10

$\begin{array}{cl}
= & \dfrac{1}{2} \times 1 \\
= & 0.5 \text{ cm}
\end{array}$

Therefore, the least possible length of the metal wire

$\begin{array}{cl}
= & 5 – 0.5 \\
= & 4.5 \text{ cm}
\end{array}$

1. Note that the maximum absolute error of the measurement

   $\begin{array}{cl}
   = & \dfrac{1}{2} \times 0.1 \\
   = & 0.05 \text{ m}
   \end{array}$

   Therefore, the maximum possible length of the thin metal wire

   $\begin{array}{cl}
   = & 2.0 + 0.05 \\
   = & 2.05 \text{ m} \\
   = & 205\text{ cm}
   \end{array}$

   Therefore, it is not possible that the actual length of this metal wire exceeds $206\text{ cm}$.

2. The maximum number of pieces of shorter metal wires

   $\begin{array}{cl}
   = & 205 \div 4.5 \\
   = & 45\dfrac{5}{9}
   \end{array}$

   Therefore, it is not possible to cut the metal wire into $46$ pieces of shorter metal wire, with each length measured as $5\text{ cm}$ correct to the neared $\text{cm}$.