$(30$ pts$)$ A power plant has a capacity of $500MW$ and an efficiency of $33\%.$ A portion of the waste heat, specifically $15\%,$ is emitted into the atmosphere through the stack, while the remaining $85\%$ is transferred to the cooling tower. The cooling water is sourced from a river and is completely evaporated in the process. Assuming the temperature of the cooling water from the river is $15C,$ please answer the following questions.

$($a$)(5$ pts$)$ Considering the efficiency of the power plant, how much is the heat input?

$($b$)(10$ pts$)$ What is the amount of waste heat taken away in the cooling water?

$($c$)(15$ pts$)$ What is the required flow rate of cooling water into the cooling tower, given that the heat of vaporization of $15C$ water is $2465k\frac{J}{k}g$ and the density of $15C$ water is $1000k\frac{g}{m^3}?$

Please solve $($b$)$