2020 MI P1 Q11

Arthur uses a special type of clay to make a multi-layered structure in which each layer is in the shape of a cylinder. The base layer has a radius of \(60\) cm and a height of \(20\) cm. The radius of each layer above is \(3\) cm less than the radius of the previous layer. The height of each layer remains the same.

1. Show that the maximum number of layers that a structure can have is \(20\).[2]
2. Ribbon is pasted around the circumference of each layer of a structure. Find the total length of a ribbon required for a \(20\)-layered structure.[3]
3. Show the total volume of a \(20\)-layered structure is given by
   \(p\pi {{\sum\limits_{n=1}^{20}{\left[ \mathrm{f}(n) \right]}}^{2}}\)
   where \(p\) is a constant to be determined and the function \(\mathrm{f}(n)\) is to be found.[3]
   Given that Arthur has \(1500\,000\) cm\(^{3}\) of clay, explain if Arthur is able to construct a \(20\)-layered structure.[1]

To make a multi-layered structure more appealing, the height of each layer is now varied such that the height of the layer above is \(80\%\) of the height of the previous layer.

4. Assuming that the height of the base layer is still \(20\) cm, determine the maximum number of layers a structure can have if the height of the structure is to be at most \(85\) cm.[3]