17. Applications of Differential Equations

Homework

1. The half-life of plutonium, \(^{236}\)Pu, is \(2.858\,\text{yr}\). How long will it take for a \(1\,\text{kg}\) sample to decay to \(1\,\text{g}\)?

2. A petri dish starts off with a culture of \(B(0)=1000\) bacteria. After \(10\,\text{hrs}\) there are \(B(10)=20\,000\) bacteria. Assuming there is no limit on resources, how many bacteria will be in the dish after \(30\,\text{hrs}\)? Plot the solution.

3. A petri dish starts off with a culture of \(B(0)=1000\) bacteria. After \(10\,\text{hrs}\) there are \(B(10)=20\,000\) bacteria. Assuming the petri dish only has enough resources to support \(M=1\,000\,000\) bacteria, how many bacteria will be in the dish after \(20\,\text{hrs}\)? After \(30\,\text{hrs}\)? After \(40\,\text{hrs}\)? Plot the solution.
   Hint: Write the logistic equation as \(\dfrac{B}{M-B}=Ae^{Mkt}\) and solve for \(A\) and \(Mk\).

4. According to Wikipedia, a soft boiled egg should be cooked to \(149^\circ\text{F}\). Several eggs are taken out of the refrigerator at \(40^\circ\text{F}\) and put in a pot of boiling water at \(212^\circ\text{F}\). One egg is removed after \(t=6\,\text{min}\) and its temperature is measured to be \(130^\circ\text{F}\). How long should the remaining eggs be cooked to be perfectly soft boiled?

5. A desalination plant has a \(1000\,\text{m}^3\) tank of salt water. At midnight, the concentration is \(3\,\dfrac{\text{kg}}{\text{m}^3}\). Salt water with concentration \(5\,\dfrac{\text{kg}}{\text{m}^3}\) is entering the tank at \(30\,\dfrac{\text{m}^3}{\text{hr}}\). Pure water is boiling off at \(10\,\dfrac{\text{m}^3}{\text{hr}}\). The water in the tank is kept thoroughly mixed and flows out an exit faucet at \(20\,\dfrac{\text{m}^3}{\text{hr}}\). When does the concentration in the tank reach \(4\,\dfrac{\text{kg}}{\text{m}^3}\)?
   Procedure: Be sure to define \(\rule{0pt}{12pt}S(t)\) as an amount not a concentration. Write out the differential equation and initial condition satisfied by \(S(t)\). Solve the equation. Using \(S(t)\), find when the concentration reaches \(4\).

6. A robot car moves so that its velocity is \(v(t)=\dfrac{1}{x(t)}\,\dfrac{\text{m}}{\text{min}}\). If the car starts at \(x(0)=2\,\text{m}\), find where the car is at \(t=6\,\text{min}\).

7. A \(0.4\,\text{kg}\) soccer ball is dropped from the top of the Sears Tower at about \(450\,\text{m}\). Assume the only force acting on the ball is acceleration due to gravity.

   1. Find its velocity, \(v(t)\), and its height, \(y(t)\), at time \(t\,\text{sec}\). Plot the height and velocity using a computer system, Maple, Mathematica, Desmos, Geogebra, etc. Say which one.

   2. When will it hit the ground? What is its velocity when it hits the ground?

8. A \(0.4\,\text{kg}\) soccer ball is dropped from the top of the Sears Tower at about \(450\,\text{m}\). Now assume air resistance with a drag coefficient of \(k=.2\,\dfrac{\text{kg}}{\text{sec}}\).

   1. Find its velocity, \(v(t)\), and its height, \(y(t)\), at time \(t\,\text{sec}\). Plot the height and velocity using a computer system. Say which one.

   2. When will it hit the ground? What is its velocity when it hits the ground?

   3. What is the terminal velocity?