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Calculus, EXT1 C1 EQ-Bank 12

A tank is initially full. It is drained so that at time `t` seconds the volume of water, `V`, in litres, is given by

`V = 50(1 - t/80)^2`  for  `0 <= t <= 100`

  1.  How much water was initially in the tank?  (1 mark)

  2.  After how many seconds was the tank one-quarter full?  (1 mark)

  3.  At what rate was the water draining out the tank when it was one-quarter full?  (2 marks)

Show Answers Only
  1.  `50\ text(L)`
  2.  `40\ text(seconds)`
  3.  `5/8\ text(litres per second)`
Show Worked Solution

i.   `text(Initially,)\ \ t = 0.`

`V` `= 50(1 – 0)^2`
  `= 50\ text(L)`

 

ii.   `text(Find)\ t\ text(when tank is)\ \ 1/4\ \ text(full:)`

`50/4` `= 50(1 – t/80)^2`
`(1 – 1/80)^2` `= 1/4`
`1 – t/80` `= 1/2`
`t/80` `= 1/2`
`t` `= 40\ text(seconds)`

 

iii.   `(dv)/(dt)` `= 2 · 50(1 – t/80)(−1/80)`
  `= −5/4(1 – t/80)`

 
`text(When)\ \ t = 40,`

`(dv)/(dt)` `= −5/4(1 – 40/80)`
  `= −5/8`

 
`:. text(Water is draining out at)\ \ 5/8\ \ text(litres per second.)`