Shown below is the structure of methyl acetate with three different...

In methyl acetate, the $\alpha$ bond is a $C=O$ double bond, which is the shortest among the three.

The $\beta$ bond is a $C-O$ single bond adjacent to a carbonyl group. Due to resonance ($CH_3-C(=O)-O-CH_3 \leftrightarrow CH_3-C(-O^-)=O^+-CH_3$), the $\beta$ bond acquires partial double bond character, making it shorter than a pure single bond.

The $\gamma$ bond is a pure $C-O$ single bond between an $sp^3$ hybridized carbon and oxygen, with no resonance involvement. Thus, it is the longest bond.

Therefore, the correct order of bond lengths is $\alpha < \beta < \gamma$.

Answer: $\alpha < \beta < \gamma$