Exponential decay

A tank contains 55 liters of water. Water is flowing out at the rate of 7% per minute. How long does it take to drain the tank?

Result

t = min (Correct answer is: INF)
t₂ =  14.2857 min

Solution:

$$\begin{gathered} p_1=7\mathrm{\%}={\displaystyle \frac{7}{100}}=0.07 \\ q=1-p_1=1-0.07={\displaystyle \frac{93}{100}}=0.93 \\ {V}_1=55\text{ l } \\ {V}_2=V_1\cdot q=55\cdot 0.93={\displaystyle \frac{1023}{20}}=51.15\text{ l } \\ {V}_3=V_2\cdot q=51.15\cdot 0.93=47.5695\text{ l } \\ {V}_4=V_3\cdot q=47.5695\cdot 0.93\doteq 44.2396\text{ l } \\ {V}_5=V_4\cdot q=44.2396\cdot 0.93\doteq 41.1429\text{ l } \\ \dots \\ V(n)=V(n-1)\cdot q \\ V(n)=V_1\cdot e^{-t\mathrm{/}\tau } \\ q=e^{-t\mathrm{/}\tau } \\ \tau =-1\mathrm{/}\ln (q)=-1\mathrm{/}\ln (0.93)\doteq 13.7797\text{ min } \\ {V}_n\mathrm{\ne }0 \\ t=\mathrm{\infty } \end{gathered}$$

$t_2=1\mathrm{/}{p}_1=1\mathrm{/}0.07={\displaystyle \frac{100}{7}}=14.2857\text{ min}$

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Showing 3 comments:

Matematik

t = infinity time to drain at rate 7% per minute of remaining volume. (exponential decay curve never touch  or cross zero - line y=0).  Time constant τ = 13.77 min

t2 = 14.286 min if we took 7% as linear volume decay, but this is not correct in this case.

see more on https://en.wikipedia.org/wiki/Exponential_decay

6 months ago  Like

Math Student

Thank you so much for your assistance.
I tried to solve using geometric sequence.
At t=~13.77 mins, there's ~20.2 liters of water left in the tank.

I do realize that the exponential function will never be zero. However, logically there must be a finite time when the tank will be emptied.
Still can't resolve the situation.

However, thank you again for your time and effort.

6 months ago  Like

Matematik

Yes, the right answer is then: the tank is empty in infinity time... T=13.77 mins is time constant, all decays can be normalized in time to have the same curve, but technically emptied at a time approx. 3-5 time constants. After time constant 13.77 mins the tank is on 63%... etc

https://en.wikipedia.org/wiki/Exponential_decay

https://en.wikipedia.org/wiki/Time_constant

6 months ago  Like

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