2013 ACJC P1 Q11

In a yogurt-making process, Jane adds \(10\) g of yogurt at \(12 \, 00\) hrs into \(500\) g of warm milk to turn the milk into yogurt. After \(t\) hours, the amount of yogurt in the mixture is \(x\) g. In a simple model, it is assumed that the rate of growth of yogurt is proportional to the product of \(x\) and \(500-x\). Show that \(x=\frac{500}{1+49{{e}^{-500kt}}}\), where \(k\) is a constant. [6]

It is given that initially, the rate of growth of yogurt is \(\frac{98}{5}\ln 2\) g per hr. Jane wishes to place the yogurt formed into the refrigerator once there is \(450\) g of yogurt in the mixture. At what time should she place the yogurt in the refrigerator?[3]