Question

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.

Answer 1

@IrishBoy123

Answer 2

@AlexandervonHumboldt2

Answer 3

so far i understand that
\[\frac{ dV }{ dt }\]
which is the water leaking is proportional to the depth...How can i mathematically express this?

Answer 4

\[\frac{ dV }{ dt } \propto h\]
means
\[\frac{ dV }{ dt } = k\; h\] where k is some constant

now start processing this with \(V = \pi r^2 h\) in mind

Answer 5

\[\frac{ dV }{ dt }=\pi r^2 \frac{ dh }{ dt }\]

so then right side will equal kh

Answer 6

no, LHS will be kh because \(\frac{ dV }{ dt } = k\; h\)

Answer 7

\[so \frac{ dh }{ dt }=\frac{ kh }{ \pi r^2 }\]

Answer 8

yes
\[so \frac{ dh }{ dt }=\frac{ kh }{ \pi r^2 } = const \times h\]

Answer 9

or \[\frac{dh}{dt} \propto h\]

Answer 10

ah yes! which is what we discussed at the beginning!! thanks!

in a nutshell - can you explain what it means when they say proportional

Answer 11

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

Answer 12

Great ! thanks heaps!