(Solved): At equilibrium, the concentrations of reactants and products can be predicted using the equ ...

At equilibrium, the concentrations of reactants and products can be predicted using the equilibrium constant, $K_{\mathrm{c}}$, which is a mathematical expression based on the chemical equation. For example, in the reaction $a\mathrm{A}+b\mathrm{B}?c\mathrm{C}+d\mathrm{D}$ where $a,b,c$, and $d$ are the stoichiometric coefficients, the equilibrium constant is $K_{\mathrm{c}}=\frac{[\mathrm{C}{]}^c[\mathrm{D}{]}^d}{[\mathrm{A}{]}^{\mathrm{a}}[\mathrm{B}{]}^b}$ where $[A],[B],[C]$, and $[\mathrm{D}]$ are the equilibrium concentrations. If the reaction is not at equilibrium, the quantity can still be calculated, but it is called the reaction quotient, $Q_c$, instead of the equilibrium constant, $K_{\mathrm{c}}$. $Q_{\mathrm{c}}=\frac{[\mathrm{C}{]}_{\mathrm{t}}{}^{\mathrm{c}}[\mathrm{D}{]}_{\mathrm{t}}{}^d}{[\mathrm{A}{]}_{\mathrm{t}}{}^a[\mathrm{B}{]}_{\mathrm{t}}{}^b}$ where each concentration is measured at some arbitrary time $t$. A mixture initially contains $\mathrm{A},\mathrm{B}$, and $\mathrm{C}$ in the following concentrations: $[\mathrm{A}]=0.500\mathrm{M},[\mathrm{B}]=0.600\mathrm{M}$, and $[\mathrm{C}]=0.350\mathrm{M}$. The following reaction occurs and equilibrium is established: $\mathrm{A}+2\mathrm{B}?\mathrm{C}$ At equilibrium, $[\mathrm{A}]=0.300\mathrm{M}$ and $[\mathrm{C}]=0.550\mathrm{M}$. Calculate the value of the equilibrium constant, $K_c$. Express your answer numerically.