  We demonstrate that using up to seven stellar abundance ratios can place
observational constraints on the star formation histories (SFHs) of Local Group
dSphs, using the Sculptor dSph as a test case. We use a one-zone chemical
evolution model to fit the overall abundance patterns of $\alpha$ elements
(which probe the core-collapse supernovae that occur shortly after star
formation), $s$-process elements (which probe AGB nucleosynthesis at
intermediate delay times), and iron-peak elements (which probe delayed Type Ia
supernovae). Our best-fit model indicates that Sculptor dSph has an ancient
SFH, consistent with previous estimates from deep photometry. However, we
derive a total star formation duration of $\sim0.9$ Gyr, which is shorter than
photometrically-derived SFHs. We explore the effect of various model
assumptions on our measurement and find that modifications to these assumptions
still produce relatively short SFHs of duration $\lesssim1.4$ Gyr. Our model is
also able to compare sets of predicted nucleosynthetic yields for supernovae
and AGB stars, and can provide insight into the nucleosynthesis of individual
elements in Sculptor dSph. We find that observed [Mn/Fe] and [Ni/Fe] trends are
most consistent with sub-$M_{\mathrm{Ch}}$ Type Ia supernova models, and that a
combination of "prompt" (delay times similar to core-collapse supernovae) and
"delayed" (minimum delay times $\gtrsim50$ Myr) $r$-process events may be
required to reproduce observed [Ba/Mg] and [Eu/Mg] trends.
