$(20$ points$)$ The time, $t,$ to drain a liquid from a hole in the bottom of a tank depends on

the initial fluid volume $V0,$ the initial fluid depth $h,$ the hole diameter $D,$ the fluid

viscosity $,$ and the fluid density $.$

a$.$ Rewrite this relationship in dimensionless form. Use $,D,$ and $$ as your repeating

variables.

b$.$ The behavior of the full$-$sized prototype is to be predicted with a $\frac{1}{5^{th}}$ scale model

$(5D_{\text{m}\text{o}\text{d}\text{e}\text{l}}=D_{\text{p}\text{r}\text{o}\text{t}\text{o}\text{t}\text{y}\text{p}\text{e}}).$ If the $\frac{1}{5^{th}}$ scale model drains in $5$ minutes, how long will it

take for the full$-$sized prototype to drain if flow similarity is achieved? Assume

the fluids used in the model and prototype have the same density and viscosity.