Problem $6$

The cylindrical storage tank sketched below has an internal diameter of $D=2\mathrm{.}2m.$ An inlet pipe supplies a constant flow of acetone $($density $790k\frac{g}{m^3})$ into the tank at a rate of $35\frac{L}{m}in.$ A feedback$-$controlled valve regulates the flow of acetone out of the tank through an outlet pipe. The valve is set such that, at all times, it provides a flow rate that is determined by the current depth of acetone in the tank, $H.$ The outflow rate in units of $\frac{m^3}{s}$ is equal to $0\mathrm{.}0004H,$ where $H$ is in $m.$

Assume that the tank is deep enough that the acetone does not spill out. If the tank is initially filled to a depth of $3\mathrm{.}0m,$ find

$($a$)$ the final steady$-$state depth of acetone in the tank, and

$($b$)$ the time it takes for the tank depth to change by $1\mathrm{.}1m$ from its initial value.