Chapter 4.3, Problem LC

LINKING concepts... For Individual or Group Explorations

A Logarithmic Model for Water Quality

The United Slates Geological Survey measures the quality of a water sample by using the diversity index d, given by

d = – [p₁• log₂(p₁) + P₂ • log₂(p₂) + • • • + p_n • log₂(p_n)],

where n is the number of different taxons (biological classifications) represented in the sample and p₁ through p_n are the percentages of organisms in each of the n taxons. For example, if 10% of the organisms in a water sample are E. coli and 90% are fecal coliform, then

d = –[0.1 • log₂(0.l) + 0.9 • log₂(0.9)] ≈ 0.5.

a) Find the value of d when the only organism found in a sample is the E. coli bacterium.

b) Let n = 3 in the formula and write the diversity index as a single logarithm.

c) If a water sample is found to contain 20% of its organisms of one type, 30% of another type, and 50% of a third type, then what is the diversity index for the water sample?

d) If the organisms in a water sample are equally distributed among 100 different taxons, then what is the diversity index?

e) If 99% of the organisms in a water sample are from one taxon and the other 1% are equally distributed among 99 other taxons, then what is the diversity index?

f) The diversity index can be found for populations other than organisms in a water sample. Find the diversity index for the dogs in the movie 101 Dalmatians (cartoon version).

g) Identify a population of your choice and different classifications within the population. (For example, the trees on campus can be classified as pine, maple, spruce, or elm.) Gather real data and calculate the diversity index for your population.