$(20$ points$)$ A city is considering building a nuclear power plant. Due to local constraints, a maximum energy input of $4,500MW$ can be supplied to this nuclear power plant, with an expected design efficiency of $30\%.$ Designers estimate that about $20\%$ of the waste could be released as heat into the atmosphere and $80\%$ of the waste heat consumed in the cooling process. A local river with upstream flow of $80\frac{m^3}{s}$ and temperature of $15C$ is expected to provide the water necessary for the cooling process.

a$.(10)$ Based on the given design parameters, what temperature change must be allowed in the cooling process to achieve a water flow rate of $40\frac{m^3}{s}?$

b$.(10)$ During the summer months the flow from the river is only $40\frac{m^3}{s}.$ If the summer river temperature is $18C$ and the power plant discharges $2\frac{m^3}{s}$ of cooling water at $80C,$ what is the final river temperature after the cooling water and the river have mixed? Ignore all heat gains and losses to the environment.