Use integration by parts to find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x \mathrm {~d} x\).
(6)
Fluid flows out of a cylindrical tank with constant cross section. At time \(t\) minutes, \(t \geq 0\), the volume of fluid remaining in the tank is \(V \mathrm {~m} ^ { 3 }\). The rate at which the fluid flows, in \(\mathrm { m } ^ { 3 } \mathrm {~min} ^ { - 1 }\), is proportional to the square root of \(V\).
Show that the depth \(h\) metres of fluid in the tank satisfies the differential equation
$$\frac { \mathrm { d } h } { \mathrm {~d} t } = - k \sqrt { } h , \quad \text { where } k \text { is a positive constant. }$$
Show that the general solution of the differential equation may be written as
$$h = ( A - B t ) ^ { 2 } , \quad \text { where } A \text { and } B \text { are constants. }$$
Given that at time \(t = 0\) the depth of fluid in the tank is 1 m , and that 5 minutes later the depth of fluid has reduced to 0.5 m ,
find the time, \(T\) minutes, which it takes for the tank to empty.
Find the depth of water in the tank at time \(0.5 T\) minutes.