Problem $1.(30$ points$)$

$1.$ A centrifugal pump with an impeller of $16$ inches in diameter develops $\backslash (20\backslash$mathrm$\{$ft$\}.\backslash$mathrm$\{$lb$\}\_\{$f$\}/\backslash$mathrm$\{$lb$\}\_\{\backslash$mathrm$\{$m$\}\}\backslash )$ of head. What head would the same type of pump running at the same RPM develop if it had a $14-$inch diameter impeller?

$2.$ A centrifugal pump with an impeller of $6$ inches in diameter is running at $1800$ RPM and using $5$ HP $.$ If the same pump is slowed to $1200$ RPM what power would be required?

$3.$ A centrifugal pump with a $10-$inch diameter is operating at $3000$ RPM$.$ What is the velocity of the fluid at the impeller tip?

$4.$ The pump in part $(3)$ is pumping a liquid with $\backslash (\backslash$mathrm$\{$SG$\}=1\mathrm{.}1\backslash ).$ What pressure in psi can the pump generate?

$5.$ A pump with a variable speed motor uses $10$ HP to pump $100$ gpm $.$ If it were slowed to pump at $2$ HP $,$ how much would it pump?