Differential Equations… does anyone know how to do these problems?
Question by Penelope: Differential Equations… does anyone know how to do these problems?
For my final, the professor handed out these questions to use as a review. He said if we know these, we should be fine on the test. My problem is, I don’t know how to do a handful of them. If you could help on them, that would be so appreciated. Thanks!
A chemical plant needed to increase the concentration of sodium chloride from 0.1% to 0.2%. The tank holds 100,000 gallons. A solution of 0.5% is pumped into the tank at 80 gallons per minute, while an equal volume is removed at the other end of the tank at the same time. Assume that there is perfect mixing in the tank.
Write out the differential equations, solve, and determine when the solution will reach 0.2%.
The population of a rare monkey is taken annually. In 1990, the population was estimated at 5000 animals, in 1995, the population was estimated at 6300 animals. Using the logistic population model, estimate the population of monkeys in 2000, 2005, and 2010.
Write out the differential equation and solve it given the conditions above
A warehouse is built without the capacity for heating or cooling. There are two insulation options. One will give the warehouse a time constant of 1 hour while the second will give a time constant of 5 hours. The outside temperature follows a sine wave given below:
Temperature(C) = 24 + 8sin(4π/3 + π t/12)
**Remember that sin (A + B) = sin(A)cos(B) + cos(A)sin(B)
where time (t) is given in 24 hour military. At t=0 (midnight on day 1), the temperature inside the warehouse is 20C
Write out the differential equation for each time constant, and determine what the temperature will be inside the ware house at 8:00 (day 1), 12:00 (day 2), and 20:00 (day 3) for each insulation package. Over the long term, describe the temperature inside the warehouse.
Use the improve Euler method with tolerance to approximate the solution to the following equation at x=1:
y’ = 1 – y + y^3, where y(0) = 0
Use a tolerance of 0.003.
This will require some programming or a spreadsheet.
Best answer:
Answer by kirchwey
I’ll take the 1st question.
dq/dt = -kq, where q = current concentration minus asymptotic concentration (exponential decay)
q(t) = q(0) * e^(-kt)
Rate constant k = 80/(60*100000) = 1.33333E-5 s^-1
(Time constant = 75000 s)
y(asymptote) = 0.005
y(0) = 0.001: q(0) = y(0)-y(asymptote)
y(t) = 0.002: q(t) = y(t)-y(asymptote)
t = ln(q(t)/q(0))/k = 21576 s
The questions I left unanswered are an hour or two of work for someone. You might consider splitting up hairy questions like these.
Know better? Leave your own answer in the comments!