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# Composite Functions Module

The two parts of this module provide practice with composite functions. They are not put in the standard format of the other Earth Math modules on this site because they are not based on real data.

## Catfish and Chemicals

A chemical factory has set up business on a river in the eastern United States. The river empties into a large lake, on which there is a restaurant that serves fresh catfish dinners. The chemical factory started dumping toxic waste into the river thus reducing the catfish population. This is described by the equation

F(w) = 50 - .5w,

where F = number of fish per acre and w = number of gallons of toxic waste dumped per day. In turn, the reduced availability of catfish caused the restaurant to increase the price of its catfish dinner. This is described by

P(F) = 24 - .4F,

where P = price of a catfish dinner.

Questions

1. Determine the composite function that describes the price of a catfish dinner in terms of the number w of gallons of toxic waste dumped by the chemical factory.
2. How much did a dinner cost before the factory opened?
3. How much waste is being dumped if the price of a dinner is \$12?
4. How much waste will kill all the catfish?

## Of Mice and Men

A deadly virus is carried by a certain kind of mouse which dwells in and around the floor of a 200 acre old growth forest in southwestern United States. On the edge of this forest is a small village with a population of 23. When the mouse population gets too large, the mice venture more into the village in search of food and new places to inhabit, thereby spreading their virus and causing villagers to become infected. There is also a certain kind of owl which dwells in this forest, and as we all know, owls love to feast on mice. Since the virus is only harmful to humans, the owls are unaffected as long as the number of owls stays at a reasonable level, the mice population is kept at a comfortable level and the virus is not transmitted to the villagers. However, visualizing large profits, the owner of the old growth forest begins to sell timber to a large lumber company; this, of course, removes the homes of the owls, and they either die off or must leave in search of other shelter.

The interrelationships of the villagers, virus, mice, owls, trees and loggers can be described by the following mathematical model.

V = number of cases of virus among villagers
M = mouse population
W = owl population
T = number of acres of trees remaining in the forest
V(M) = .025M - 20
M(W) = 1600 - 20W
W(T) = .2T

Use this model to answer these questions.

1. What is the domain and range of each of the three functions in the model?
2. What is the maximum number of owls which can survive in the forest? What is the maximum number of mice which can survive? What is the maximum number of viral cases that can occur among the villagers?
3. Determine the composite function that describes the number of mice in terms of the number of acres of timber remaining in the forest. Also, determine the composite function that describes the number of cases of viral infection among the villagers in terms of the number of acres of trees remaining in the forest.
4. How many cases of virus will occur if the forest is untouched? If half the forest is logged? If three-fourths is logged?
5. How many villagers will be left if the forest is totally logged?