Mathematics 134 – Intensive Calculus for Science 2 Discussion 5 – Growth and Decay April 24, 2002

Background

We have seen how statements about rates of change can be translated into differential equations. Providing the resulting equation is relatively simple (e.g. a separable equation), we can solve it and derive a formula for the solution. We can use that solution and other information to make predictions about how the quantities related by the differential equation will behave. Today, we will practice on several problems of this type. Note: Even though this material is not covered on the exam coming up this Friday, you will be responsible for it on the final exam.

Discussion Problems

I. Carbon-14 is a radioactive isotope of carbon that decays with a half-life of about 5370 years (the rate of change of the amount carbon-14 is proportional to the amount at all times). There is a relatively steady supply of this form of carbon in nature, produced by decay of other radioactive elements. Living organisms incorporate carbon-14 into their tissues when they ingest nutrients, but when they die, the carbon-14 stops being replenished and the amount left decreases over time. The radioactive carbon-14 in a one gram sample of carbon from living tissue from an oak tree produces about 13.5 atom disintegrations per minute. In 1977, charcoal (made by burning oak wood at some time in the past) was found near Stonehenge in England. It registered 8.2 disintegrations per minute per gram of carbon. Using this information, estimate how old the charcoal is (how long since the wood was living).

II. The population of fish in a lake is attacked by a microscopic water-borne parasite at t = 0, and as a result the population declines at a rate proportional to the square root of the population from that time on. A) Express this statement about the rate of growth of the population P as a differential

equation. B) Use separation of variables to find the formula for the general solution of your differ-

ential equation. C) At t = 0 there were 900 fish in the lake; 441 were left after 6 weeks. When did the

fish population disappear entirely?

Assignment

Group writeups due in class on Monday, April 29.