Reaction Rates In a chemical reaction, the reaction rate R is a function of the concentraton of the product of the reaction. For a certain second-order reaction between two substances, we have the formula R=0.01x2x+22. Here x is measured in moles per cubic meter and R is measured in moles per cubic meter per second. a. Make a graph of R versus x. Include concentrations up to 100 moles per cubic meter. b. Use functional notation to express the reaction rate when the concentration is 15 moles per cubic meter, and then calculate hat value. c. The reaction is said to be in equilibrium when the reaction rate is 0. At what two concentratoins is the reaction in equilibrium?

Spawner-Recruit Model In fish management it is important to know the relationship between the abundance of the spawners also called the parent stock and the abundance of the recruitsthat is, those hatchlings surviving to maturity. According to the Ricker model, the number of recruits R as a function of the number of spawners P has the form R=APeBp for some positive constants A and B. This model describes well a phenomenon observed in some fisheries: A large spawning group can actually lead to a small group of recruits. In a study of the sockeye salmon, it was determined that A=4 and B=0.7. Here we measure P and R in thousands of salmon. a. Make a graph of R versus P for the sockeye salmon. Assume there are at most 3000 spawners. b. Find the maximum number of salmon recruits possible. c. If the number of recruits R is greater than the number of spawners P, then the difference R-P of the recruits can be removed by fishing, and next season there will once again be P spawners surviving to renew the cycle. What value of P gives the maximum value of R-P, the number of fish available for removal by fishing?

5. United States Population Growth In 1960 the population of the United States was about 180 million. Since that time, the population has increased by approximately 1.2 each year. This is a verbal description of the function N=N(t), where N is the population, in millions, and t is the number of years since 1960. a. Express in functional notation the population of the United States in 1963. Calculate its value. b. Use the verbal description of N to make a table of values that shows U.S. population in millions from 1960 through 1965. c. Make a graph of U.S. population versus time. Be sure to label your graph appropriately. d. Verify that the formula 1801.012tmillion people, where t is the number of years since 1960, gives the same values as those you found in the table in part b. Note: Because t is the number of years since 1960, you would use t = 2 to get the population in 1962. e. Assuming that the population has been growing at the same percentage rate since 1960, what value does the formula above give for the population in 2000? Note: The actual population in 2000 was about 281 million..