A thin-walled vessel is produced by rotating the graph of    about the -axis for  .

All lengths are measured in centimetres.

1. 1. Write down a definite integral in terms of and for the volume of the vessel in cubic centimetres.   (1 mark)
      

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   2. Hence, find an expression for the volume of the vessel in terms of .   (1 mark)
      

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Water is poured into the vessel. However, due to a crack in the base, water leaks out at a rate proportional to the square root of the depth of water in the vessel, that is  , where is the volume of water remaining in the vessel, in cubic centimetres, after minutes.

2. 1. Show that  .   (2 marks)
      

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   2. Find the maximum rate, in centimetres per minute, at which the depth of water in the vessel decreases, correct to two decimal places, and find the corresponding depth in centimetres.   (2 marks)
      

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   3. Let    for a particular vessel. The vessel is initially full and water continues to leak out at a rate of    cm³ min.
   4. Find the maximum rate at which water can be added, in cubic centimetres per minute, without the vessel overflowing.   (1 mark)
      

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3. The vessel is initially full where    and water leaks out at a rate of    cm³ min. When the depth of the water drops to 25 cm, extra water is poured in at a rate of    cm³ min.
4. Find how long it takes for the vessel to refill completely from a depht of 25 cm. Give your answer in minutes, correct to one decimal place.   (3 marks)
   

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