An industrial chemical reactor vessel with a circular cross-section and a maximum depth of 30 cm is initially empty. A cooling reagent is pumped into the reactor such that its depth at time ttt seconds is hhh cm.
The volume of reagent in the reactor, V cm3V \text{ cm}^3V cm3, is modelled by the formula:
V=112h2(2h+45)0≤h≤30 V = \frac{1}{12}h^2(2h + 45) \quad 0 \le h \le 30 V=121h2(2h+45)0≤h≤30The reagent is delivered at a constant rate of 225 cm3 s−1225 \text{ cm}^3\text{ s}^{-1}225 cm3 s−1. According to this model:
Determine the time required to fill the reactor vessel completely.
Calculate the rate of change of the depth of the reagent, in cm s−1\text{cm s}^{-1}cm s−1, at the instant the depth reaches 15 cm.
