Differential equations assignment
I will be checking for organization, conceptual understanding, and proper mathematical communication, as well as completion of the problems.
• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing.
• Use correct mathematical notation.
• Show your work vertically.
• You may work with your classmates. However, please submit your own work!
• Work on a separate sheet of paper, and label each problem clearly.
1. Carbon Dating An important tool in archeological research is radiocarbon dating, developed by the American chemist Willard F. Libby. This is a means of determining the age of certain wood and plant remains, hence of organic materials such as animal or human bones or artifacts found buried at the same levels. Radiocarbon dating is based on the fact that some wood or plant remains contain residual amounts of carbon-14 (C-14)−A radioactive isotope of carbon. This isotope is accumulated during the lifetime of the plant and begins to decay at its death. Since the half-life (i.e., the amount of time it takes for a quantity of radioactive material to decay to one-half of its original amount) of C-14 is very long (approximately 5730 years!!!), measurable amounts of C-14 remain after many thousands of years. If even a tiny fraction of the original amount of C-14 is still present, then by appropriate laboratory measurements the proportion of the original amount of C-14 that remains can be accurately determined. In other words, if Q(t) is the amount of C-14 at time t and Q 0 is the original amount, then the ratio Q(t)/Q 0 can be determined, at least if this quantity is not too small. (FYI: The present techniques permit the use of this method for time periods of 50,000 years or more!)
(a) (3 points) Assuming that Q satisﬁes the diﬀerential equation
dQ = −rQ , dt
ﬁnd an expression for Q(t) at any time t (in years), if Q(0) = Q 0 . That is, solve the corresponding IVP.
(b) (2 point) Determine the decay constant r for C-14 to THREE signiﬁcant digits. Also, what is the unit of this decay constant?
(c) (5 points) Suppose that certain remains are discovered in which the current residual amount of C-14 is 25% of the original amount. Determine the age of these remains to the nearest year.
2. Contaminated Swimming Pool A swimming pool at a water park contains 50,000 gallons of water. Unfortunately, it has been contaminated by 3,000 g of a nontoxic dye that leaves a swimmer’s skin an unattractive green. The pool’s ﬁltering system can take water from the pool, remove the dye, and return the water to the pool at a ﬂow rate of 250 gal/min.
(a) (4 points) Write down the IVP for the ﬁltering process; let D(t) be the amount of dye (in grams) in the pool at time t (in minutes). After writing down the IVP, solve it!
(b) (2 points) The water park is scheduled to open in 3 hours. It has been determined that the eﬀect of the dye is imperceptible if its concentration is less than 0.02 g/gal. Is the ﬁltering system capable of reducing the dye concentration to this level within 3 hours?
(c) (2 points) How long would it take for the concentration of dye to ﬁrst reach the value 0.02 g/gal?
(d) (2 points) Find the ﬂow rate (to the nearest integer) that is suﬃcient to achieve the concentration 0.02 g/gal within 3 hours. Make sure to include correct units.