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:

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

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

Try another example

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

3 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.

3 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

3 months ago  Like

Next similar math problems:

• Std-deviation
  Calculate standard deviation for file: 63,65,68,69,69,72,75,76,77,79,79,80,82,83,84,88,90
• Water reservoir
  One drain hole emptied the water from the tank for 20 hours, with the second drain for 30 hours. How long will the water drain from this tank if we open two drains simultaneously?
• Pouring alcohol
  100 liters of alcohol has 70% How many liters of water need to be added to have 60% alcohol?
• The crime
  The crime rate of a certain city is increasing by exactly 7% each year. If there were 600 crimes in the year 1990 and the crime rate remains constant each year, determine the approximate number of crimes in the year 2025.
• Savings
  Suppose on your 21st birthday you begin making monthly payments of $500 into an account that pays 8% compounded monthly. If you continue the payments untill your 51st birthday (30 years), How much money will be in your account? How much of it is interest?
• Crystal
  The crystal grows every month 1.2 permille of its mass. For how many months to grow a crystal from weight 177 g to 384 g?
• To the cinema
  Jane at 8 am got message that all 1093 school pupils will go to the cinema. Within 20 min she said it to the three friends. Each of them again for 20 minutes said to the other three. In this way the message spread further. At what time all the children in
• Sequence
  Calculate what member of the sequence specified by ? has value 86.
• Alcohol 2
  Two types of alcohol one 63% and second 75% give 20 liters of 69% alcohol. How many liters of each type are in the mixture?
• Annual income
  The annual incomes (in thousands of $) of fifteen families is: 60, 80, 90, 96, 120, 150, 200, 360, 480, 520, 1060, 1200, 1450, 2500, 7200 Calculate harmonic and geometric mean.
• Hot air balloon
  Hot air balloon ascends 25 meters up for a minute after launch. Every minute ascends 75 percent of the height, which climbed in the previous minute. a) how many meters ascend six minutes after takeoff? b) what is the overall height 10 minutes after launch
• Geometric sequence
  In the geometric sequence is a₄ = 20 a₉ = -160. Calculate the first member a₁ and quotient q.
• Population growth
  How many people will be on Earth from two people for 5,000 years, if every couple has always 4 children, (2 boys and 2 girls) at the age of 25-35, and every man will live 75 years?
• Sum of series
  Determine the 6-th member and the sum of a geometric series: 5-4/1+16/5-64/25+256/125-1024/625+....
• Alcohol mixing
  How much 64% of the alcohol must be poured into 6 liters of 85% alcohol to produce 79% alcohol?
• Alcohol
  How many 55% alcohol we need to pour into 14 liters 75% alcohol to get p3% alcohol? How many 65% alcohol we get?
• Chess
  How many ways can select 4 fields on classic chess board with 64 fields, so that fields don't has the same color?