Name: Rebecca Edwards
Who is asking: Student
Level: Secondary

A tank in which cholocate milk is being mixed contains a mixture of 460 liters of milk and 40 liters of chocolate syrup initially. Syrup and milk are then added to the tank at the rate of 2 liters per minute of syrup and 8 liters of milk per minute. Simultaneously the mixture is withdrawn at the rate of 10 liters per minute. Find the function giving the amount of syrup in the tank at time t.

-Ok, I haven't got very far with this problem because its complicated. I know that I should start by finding differential equations, especially for the rate of change of the syrup, but I'm confused because its being added and subracted at the same time. I also know the volume stays constant, at 500 liters, because the total amount being added and subtracted are the same: 10 liters per minute. But the ratios are different and I don't know how to figure it out. Please help! Thank you!

Hi Rebecca,

It helps here to think in terms of input and output. If the amount of syrup in the tank at time t is s(t) then

ds/dt = syrup input rate - syrup output rate

The syrup input rate you know, it is 2 liters/minute.

The liquid output rate is 10 liters/minute and the proportion of the liquid that is syrup is s(t)/500, hence the syrup output rate is s(t)/500 x 10 liters/minute. Thus

ds/dt = 2 - s(t)/50 liters/minute.

This is the differential equation you need to solve.

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