$(20$ PTS$.)$ A tank in the shape of a paraboloid of revolution is completely filled with water and drained via an orifice at the bottom with orifice area, $A_o$ and orifice discharge coefficient $C_d.$

Determine:

a$.$ The correct relationship of the drain time of the tank as a function of $R,H,A_o,C_d$ and $g$

b$.$ the time $($sec$)$ required to completely drain the full tank for $H=10ft,R=3ft,C_d=0\mathrm{.}1,A_o=3i{n}^2,g=32\mathrm{.}2f\frac{t}{s^2}$

HINTS:

From the geometry: $\frac{r^2}{R^2}=\frac{h}{H}$

Volume of paraboloid: $AA=\frac{r^{2}h}{2}$

$(20$ PTS$.)$ Water flows out of the sprinkler pipe via a $\frac{1}{8}$ inch slot as shown where $=1\mathrm{.}94$ slugs$/f{t}^3.$

The total flow rate into the sprinkler is is $1f\frac{t^3}{sec}$ and the exit velocity varies linearly as shown. Determine:

$($a$)$ The correct expression for the velocity distribution in the slit in the form $u(x)=mx+b$ where $m$&$b$ are constants

$($b$)$ The moment, $T($ft$-$lbf$)$ about the axis of rotation $M-M$ of the sprinkler set up by the angular momentum of the water exiting the slot

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