A hemispherical water tank has radius cm. The tank has a hole at the bottom which allows water to drain out.

Initially the tank is empty. Water is poured into the tank at a constant rate of  cm³ s, where is a positive constant.

After seconds, the height of the water in the tank is cm, as shown in the diagram, and the volume of water in the tank is cm³.
  

It is known that      (Do NOT prove this.)

While water flows into the tank and also drains out of the bottom, the rate of change of the volume of water in the tank is given by  .

1. Show that  .  (2 marks)
   

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2. Show that the tank is full of water after  seconds.  (2 marks)
   

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3. The instant the tank is full, water stops flowing into the tank, but it continues to drain out of the hole at the bottom as before.
4. Show that the tank takes 3 times as long to empty as it did to fill.  (3 marks)
   

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