Bakshi problem $4\mathrm{.}5($modified$)$: In a free market, the marginal cost $(M)$ of a commodity in $\frac{\text{\$}}{k}g$ may be modeled as$,M=3+\frac{Q^2}{2},$ where $Q$ is the quantity produced in kilograms. The demand for this commodity is related to its price $(P)$ as$,P=120-2\mathrm{.}5Q.$ Answer the following questions,

a$.$ Determine the equilibrium point for this market.

b$.$ If an environmental tax of $$\frac{50}{k}g$ is imposed on this system, what is the new equilibrium point?