marigirl
  • marigirl
Could someone please help me understand this question and specifically what it means by proportional: Water leaks from the bottom of a tank at a rate proportional to the depth h of the water in the tank. Write down an equation describing this if V is the volume of the water in the tank after time 't' and if the tank is a cylinder show that dh/dt is proportional to h.
Mathematics
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jamiebookeater
  • jamiebookeater
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marigirl
  • marigirl
@IrishBoy123
marigirl
  • marigirl
@AlexandervonHumboldt2
marigirl
  • marigirl
so far i understand that \[\frac{ dV }{ dt }\] which is the water leaking is proportional to the depth...How can i mathematically express this?

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IrishBoy123
  • IrishBoy123
\[\frac{ dV }{ dt } \propto h\] means \[\frac{ dV }{ dt } = k\; h\] where k is some constant now start processing this with \(V = \pi r^2 h\) in mind
marigirl
  • marigirl
\[\frac{ dV }{ dt }=\pi r^2 \frac{ dh }{ dt }\] so then right side will equal kh
IrishBoy123
  • IrishBoy123
no, LHS will be kh because \(\frac{ dV }{ dt } = k\; h\)
marigirl
  • marigirl
\[so \frac{ dh }{ dt }=\frac{ kh }{ \pi r^2 }\]
IrishBoy123
  • IrishBoy123
yes \[so \frac{ dh }{ dt }=\frac{ kh }{ \pi r^2 } = const \times h\]
IrishBoy123
  • IrishBoy123
or \[\frac{dh}{dt} \propto h\]
marigirl
  • marigirl
ah yes! which is what we discussed at the beginning!! thanks! in a nutshell - can you explain what it means when they say proportional
IrishBoy123
  • IrishBoy123
for proportional think straight lines, eg \(y = mx + x\), which means \(y \propto x\) the relationship is linear, as x increases y increases. contrast with inversely proportional. here think \(y = \frac{1}{x}\), so \(y \propto \frac{1}{x}\), as y is "inversely proportional" to x
marigirl
  • marigirl
Great ! thanks heaps!

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