A distribution network supplies demand Q$1,$ Q$2,$ and Q$3$ at three locations $($in L$/$s$).$ The

relationship between pressure head at node $2$ of the network and the three demands is

explained by the following function:

$h2=f(Q_1,Q_2,Q_3)=112m-0\mathrm{.}001Q_1-0\mathrm{.}003Q_2^2-0\mathrm{.}006Q_3^4$

where h$2=$ pressure head at node $2($in metres$).$ The mean and variance of the

demands are indicated in Table $1.$ The demands are statistically independent $($no

correlation$).$

Table $1.$ Mean and variance of $3$ demands in the distribution network.

a$)$ Use the First Order Second Moment $($FOSM$)$ method to calculate the mean and

standard deviation of pressure head at node $2.$

b$)$ If all the demands had the same variance, which would contribute the most to the

value of standard deviation of pressure head at node $2?$ Why?

c$)$ If the variance of demands Q$1$ and Q$2$ is increased, will the standard deviation of

pressure head at node $2$ increase or decrease? Explain.

d$)$ If the $3$ demands Q$1-$Q$3$ are highly correlated in space, will that increase or decrease

the standard deviation of pressure head at node $2?$ Explain.