Here's one type that's very common to do with rates and concentrations:
A 100 gallon tank is full of pure water. Let pure water run into the tank at the rate of 2 gals/ min. and a brine solution containing 1/2 lb. of salt run in at the rate of 2 gals/min. The mixture flows out of the tank through an outlet tube at the rate of 4 gals/min. Assuming perfect mixing, what is the amount of salt in the tank after t minutes?
Let s be the amount of salt in the tank in pounds at time t. Then:
s/ 100 = concentration of salt (i.e. as a proportion of total gallons of pure water in tank initially)
Then: ds/ dt = net rate of change = (rate of gain in lbs/min - rate of loss in lbs/min)
We can further write:
ds/dt = 1 - 4s/ 100 = 1 - s/25
Writing the basic differential equation to solve: ds/ (25 - s) =
This requires integrating both sides:
Which is the basic equation for a simple harmonic oscillator:
w = 2 pf so:
The total force acting is therefore:
F = 100.2 - 40.2 - 2v = 60 - 2v
WHY is this? We have two negative contributions (40.2 and 2v) on the LHS because the force of friction and drag both act opposite to the direction of motion. '2v' because the drag is stated as 'twice the velocity'. The starting DE becomes:
(1000)/ g (dv/dt) = 60- 2v
Re-arranging to separate variables:
dv / (30 - v) = (32.17) dt/ 500 = 0.06434 dt
For which the solution is obtained by integration, i.e.
ln (30 - v) - ln 30 = -0.06434 t
Taking natural logs of each side:
(30 - v)/ 30 = e –0.06434t
v = 30 (1 - e –0.06434t )
After 10 seconds: v = 30 (1 - e –0.6434 ) = 30 (1 -0.5255) = 30 (0.4745)
v(10) = 14.24 feet/sec
Problems for the Math Maven:
1) A ship weighing 64,000 tons starts from rest under the impetus of a constant propeller thrust of 200,000 lbs. The resistance of the water is 10,000v lbs.
Find the velocity v as a function of the time and the terminal velocity in miles per hour (let g = 32 f/s/s)
2) A 5 lb. weight hangs vertically on a spring whose spring constant k = 10. The weight is pulled 6 inches farther down and released. Find the equation of motion, its period and the frequency. What would the units be for k?
3) A tank contains 200 gallons of brine with 100 pounds of dissolved salt in solution. Salt water containing 1 pound per salt per of gallon water enters the tank at the rate of 5 gallons per minute and the brine flows out at the same rate.
Find the amount of salt left in the tank at the end of one hour.