Combine a leaking cone with trapezium optimisation

(a) A factory produces disposable cups in the shape of inverted right cones of radius $4$ cm and height $8$ cm. For a quality check, the supervisor fills one cup completely with water. Water leaks from its vertex at a constant $1.5\text{ cm}^3\text{ s}^{-1}$. Find the exact rate at which the water level is decreasing when its depth is $2$ cm.4 marks

The volume of a cone of base radius $r$ and height $h$ is $\tfrac13\pi r^2h$.

(b) Trapezium $ABCD$ has height $5$ cm and $DA+AB+BC=20$ cm. Points $M,N$ are the feet of the perpendiculars from $A,B$ to $DC$. Given $\angle DAM=\angle CBN=\theta$, where $0<\theta<\pi/2$:

(i) Show that its area is $A=100-50\sec\theta+25\tan\theta\text{ cm}^2$.3 marks

(ii) Hence use differentiation to find its maximum area in exact form. You do not need to verify that it is a maximum.4 marks