N2022 P2 Q2

N2022 P2 Q2

Junior College 2
7 marks
Free

A cylindrical container has a base area of \(0.9\ \mathrm{m}^2\) and height \(h\ \mathrm{m}\). Water is poured into the container at the rate of \(k\) litres per second. The container is filled in 72 seconds.

[\(1\ \mathrm{m}^3=1000\) litres]

  1. Show that \(k=12.5h\).[1]

The container is emptied. Water is now poured into the container at the rate of \(kt\) litres per second, where \(t\) is the time in seconds from when pouring begins.

  1. Find the time taken to fill the container.[3]

The container is emptied again. Water is now poured into the container at the rate of \((kt+25)\) litres per second, where \(t\) is the time in seconds from when pouring begins. The container is filled in 10 seconds.

  1. Find the height of the container, \(h\).[3]

Default solution

  1. \(V=0.9h\ \mathrm{m}^3=900h\) litres
    \(k=\frac{900h}{72}=12.5h\) (shown)
  2. \(\frac{\mathrm{d}V}{\mathrm{d}t}=kt=12.5ht\)
    \(V(t)=\frac{25}{4}ht^2\), since \(V(0)=0\).
    \(900h=\frac{25}{4}ht^2\Rightarrow t^2=144\Rightarrow t=12\ \mathrm{s}\) (\(t>0\))
  3. \(\frac{\mathrm{d}V}{\mathrm{d}t}=kt+25=12.5ht+25\)
    \(V(t)=\frac{25}{4}ht^2+25t\), since \(V(0)=0\).
    \(900h=\frac{25}{4}h(10)^2+25(10)=625h+250\)
    \(h=\frac{250}{275}=\frac{10}{11}\ \mathrm{m}\)
Answer:(a) \(k=12.5h\) (b) \(12\ \mathrm{s}\) (c) \(h=\frac{10}{11}\ \mathrm{m}\)

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