Differentiation - Connected Rates of Change

A square-based prism fish tank with no lid has a fixed volume of \(2000\text{ cm}^3\). The cost of the materials to manufacture this fish tank is such that, the cost of the base per \(\text{cm}^2\) is twice the cost of the side faces per \(\text{cm}^2\). Assume that the side faces cost \(\$ 1\) per \(\text{cm}^2\) and that the thickness of the tank is negligible. Let \(x\text{ cm}\) be the length of the side of the base square.

1. Find the value of \(x\) for which the total cost of manufacturing the fish tank is a minimum.

Another fish tank of different shape is initially empty. Water is pumped in the fish tank at a constant rate of \(4\text{ cm}^3/\text{s}\). Let \(h\text{ cm}\) be the depth and \(V\text{ cm}^3\) be the volume of water in the tank at any time. It is given that \(V = (2h^2 + h)^{\frac{2}{3}}\).

2. Find the rate of change of \(h\) when \(h = 10\text{ cm}\).