Lactobacillus casei is propagated under essentially anaerobic conditions to provide a starter culture for the manufacture of Swiss cheese. The culture produces lactic acid as a by-product of energy metabolism. The system has the following characteristics:                                                                  Yxs = = 0.25 kg kg-1

  Ks = 0.14 kgm-3
   μmax = 0.30 h-1
  ms = 0.132 kg kg-1 h-1

A stirred fermenter is operated in fed-batch mode at quasi- steady-state with a feed flow rate of 5 m3 h-1 and feed substrate concentration of 80 kg m-3. After 6 h, the liquid volume is 50 m3. 

(a) What was the initial culture volume?
(b) What is the concentration of substrate at a quasi-steady state?
(c) What is the concentration of cells at a quasi-steady state?
(d) What mass of cells is produced after a 6 h fed-batch operation?