Consider the following molecules/species: (x) tropone, (y) acetone,...

In acetone $(y)$, the carbon-oxygen bond is a localized double bond with a bond order of approximately $2$.

In the acetate ion $(z)$, there are two equivalent resonance structures that delocalize the negative charge equally over both oxygen atoms. This results in a carbon-oxygen bond order of exactly $1.5$.

In tropone $(x)$, the carbon-oxygen double bond is conjugated with the $\pi$ system of the seven-membered ring. Polarization of the $C=O$ bond leads to a resonance structure with a single $C-O^{-}$ bond and a positive charge on the ring, forming an aromatic tropylium cation. This aromatic stabilization gives the $C=O$ bond significant single-bond character. However, because this resonance structure involves charge separation, its contribution is less than that of the neutral structure. Thus, the bond order in tropone is between $1.5$ and $2$.

Since bond length is inversely proportional to bond order, the carbon-oxygen bond length is longest for the acetate ion (bond order $1.5$), followed by tropone (bond order between $1.5$ and $2$), and shortest for acetone (bond order $2$).

Therefore, the correct order of carbon-oxygen bond length is $z > x > y$.

Answer: $z>x>y$