2023 TJC P1 Q10

Water is being pumped from reservoir into a filtration device at a constant rate of \(8\) gallons per hour. The filtration device processes the water and discharges it at a rate proportional to the volume of water currently in it. At time \(t\) hours, the volume of water in the device is \(v\) gallons. The filtration device is initially empty. After \(10\) hours, the volume of water in the device is \(5\) gallons.

1. Find an expression for \(v\) in terms of \(t\).[7]
2. Hence find the volume of water in the filtration device in the long run.[1]

A fault in the filtration device is discovered when there are \(4.5\) gallons of water in the device. The device now discharges water at a rate that is equal to the square of the volume of water currently in it.

3. Write down a differential equation to model the new situation.[1]
4. Find the time (to the nearest minute) that elapsed from the moment the fault is discovered to the moment when the water in the device drops to \(3\) gallons.[4]