The figure below shows a container with an open top. The uniform cross section $A B C L$ of the container is a trapezium with $A B=B C=C D=10 \mathrm{~cm} . A B$ and $C D$ are eacl inclined to the line $B C$ at an acute angle of $\theta$ radians.
The length of the container is $50 \mathrm{~cm}$ and the container is placed on a horizontal table.

(i) Show that the volume $V$ of the container is given by
$V=5000(\sin \theta)(1+\cos \theta) \mathrm{cm}^3$


Hence using differentiation, find the exact maximum value of $V$, proving that it is a maximum.


(ii) For the remaining part of the question, $\theta$ is fixed at $\frac{\pi}{4}$.
Water fills the container at a rate of $100 \mathrm{~cm}^3 \mathrm{~s}^{-1}$. At time $t$ seconds, the depth of the water is $h \mathrm{~cm}$. The surface of the water is a rectangle $P Q R S$. When $h=3 \mathrm{~cm}$, find the

(a) the depth of the water, $h$,


(b) the surface area of the water $P Q R S$.