Differentiation - Problems Involving Maximisation and Minimisation

A factory manufactures cylindrical containers with fixed volume of \(\frac{5\pi}{128}\) cm\(^3\). The material used for the top and the base of the cylinder costs \(\$ \ a\) / cm\(^2\) and the material used for the curved surface costs \(\$ \ \frac{4a}{5}\) / cm\(^2\). Assume that the thickness of the container is negligible.

1. Given that the container has a base radius of \(r\) cm, which can vary, find the total cost of manufacturing one such container in terms of \(r\).
2. Hence find the value of \(r\) for which the total cost has a minimum value.