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Milk chocolate is being produced in a 300 litre tank, which initially contains 100 litres of milk. The following things then occur simultaneously:

1. Liquid cocoa (made up of equal parts cocoa solids and cocoa butter, both in liquid form) is added at a rate of 6 litres per minute.

2. Milk is added at a rate of 3 litres per minute. The well-stirred mixture leaves the tank via a tube, at a rate of 6 litres per minute.

Let x(t) be the amount of cocoa solids in the mixture

The differential equation is dx/dt + 6x / (100 + 3t) = 3

......Therefore......

x(t) = (9t^3 + 900t^2 + 30,000t) / (3t + 100)^2

After the tank is full, the process continues as above. However, in addition to the well-stirred mixture leaving via the tube, it also flows over the edges of the tank and is collected by overflow tubing which takes it to the cooling process. Let y(t) be the number of litres of cocoa solids present in the tank t minutes after it is full.

Find the differential equation satisfied by y(t)

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# Differential Equations - Mixture in an overflowing tank

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