You are a production engineer at ConCo, responsible for a production plant that packages concrete

mix. The mix comes in bags of $50$ kg$,$ and consist of $50$ weight $\%$ cement, $25$ weight $\%$ sand, and

$25$ weight $\%$ gravel. A bag of concrete mix is filled by sequentially adding the three components

to the bag, and then mixing it$.$ The weight of each component added to the bag is normally

distributed, with mean $25$ kg and standard deviation $0\mathrm{.}2$ kg for cement, mean $12\mathrm{.}5$ kg and standard

deviation $0\mathrm{.}2$ kg for sand, and mean $12\mathrm{.}5$ kg and standard deviation $0\mathrm{.}4$ kg for gravel. The

correlation coefficient between the weights of any two materials is $0\mathrm{.}5.$

a$)$ What is the mean and standard deviation of the weight of the bags of concrete mix?