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 =  14.286 min

Solution:

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

$t_2=1\mathrm{/}{p}_1=1\mathrm{/}0.07={\displaystyle \frac{100}{7}}=14.286\text{ min}$

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

25 days 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.

25 days 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

25 days ago  Like

Next similar math problems:

• 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.
• Population
  The town has 65,000 inhabitants. 40 years ago there were 157,000. How many people will live in town in 10 years if the average rate in population is as in previous years?
• Population
  What is the population of the city with 3% annual growth, if in 10 years the city will have 60,000 residents?
• Radioactive material
  A radioactive material loses 10% of its mass each year. What proportion will be left there after n=6 years?
• Investment
  1000$ is invested at 10% compound interest. What factor is the capital multiplied by each year? How much will be there after n=12 years?
• Radioactivity
  After 548 hours decreases the activity of a radioactive substance to 1/9 of the initial value. What is the half-life of the substance?
• Insert into GP
  Between numbers 5 and 640 insert as many numbers to form geometric progression so sum of the numbers you entered will be 630. How many numbers you must insert?
• GP - 8 items
  Determine the first eight members of a geometric progression if a₉=512, q=2
• Five members
  Write first 5 members geometric sequence and determine whether it is increasing or decreasing: a₁ = 3 q = -2
• Tenth member
  Calculate the tenth member of geometric sequence when given: a₁=1/2 and q=2
• Researchers
  Researchers ask 200 families whether or not they were the homeowner and how many cars they had. Their response was homeowner: 14 no car or one car, two or more cars 86, not homeowner: 38 no car or one car, two or more cars 62. What percent of the families
• A perineum
  A perineum string is 10% shorter than its original string. The first string is 24, what is the 9th string or term?
• Simple interest 3
  Find the simple interest if 11928 USD at 2% for 10 weeks.
• Six terms
  Find the first six terms of the sequence a1 = -3, an = 2 * an-1
• Five harvests
  In the seed company, they know that, out of 100 grains of a new variety, they get an average of 2000 grains after harvest. Approximately how many grains do they get out of 100 grains after five harvests?
• Geometric progression 2
  There is geometric sequence with a₁=5.7 and quotient q=-2.5. Calculate a₁₇.
• Sequence
  Find the common ratio of the sequence -3, -1.5, -0.75, -0.375, -0.1875. Ratio write as decimal number rounded to tenth.