## marigirl one year ago 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.

1. marigirl

@IrishBoy123

2. marigirl

@AlexandervonHumboldt2

3. 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?

4. 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

5. marigirl

$\frac{ dV }{ dt }=\pi r^2 \frac{ dh }{ dt }$ so then right side will equal kh

6. IrishBoy123

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

7. marigirl

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

8. IrishBoy123

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

9. IrishBoy123

or $\frac{dh}{dt} \propto h$

10. 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

11. 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

12. marigirl

Great ! thanks heaps!