A thermos flask is modelled by a cylinder and a hemisphere, as shown in the diagram.
The cylinder has a radius of \(5x\) cm and a height of \(y\) cm.
The hemisphere has a radius of \(4x\) cm.
Show that the total volume, \(V\), of the flask, in terms of \(x,y\) and \(\pi\) is \(V=25\pi x^2y+\dfrac{128}{3}\pi x^3\).[2]
Given the total volume is \(2700\pi\) and \(x=1.5\), find the value of \(y\).[2]
The flask is covered with a layer of insulating material over the entire external surfaces, including the flat base. The insulating material costs $0.002 per \(\text{cm}^2\).
Using your answer in part (b), calculate the total cost of covering the flask.[3]