Advanced Algebra
Chapter 9 Answers
Exponential and Logarithmic Functions
Exponential functions can be used to model the growth and decline of populations of endangered species.
Part B, Exponential Growth and Decay
1. Read about population changes that the bald eagle, America's national bird, has undergone during the past 200 years.
a. Give estimated numbers of bald eagles in the United States in 1800, 1960, and 1995. (Note that the given data are for pairs of eagles. [150,000; 900; 9000]
b. Calculate the average annual rate of decline in the bald eagle population from 1800 to 1960. Calculate the average annual rate of increase from 1960 to 1995. [3.15% annual decrease; 6.8% annual increase]
c. Write and graph equations modeling the population decline (1800-1960) and population increase (1960-1995). [decline P = 150,000(0.9685)t]
[increase P = 900(1.068)t]
d. If the population decline had not been halted, when would the bald eagle population have fallen below 10 birds (5 pairs)? Without intervention, when would have the species become extinct? [about 300 years after 1800, or 2100]
e. If the current rate of increase continues, when will the population exceed the 1800 population? [about 78 years after 1960, or 2038]
Part C, Modeling Exponential Growth and Decay
2. Like the bald eagle before it rebounded, the whooping crane is a gravely endangered species.
a. Collect and graph data on whooping crane numbers, 1940-1996.
b. Use exponential regression to find an exponential model y = abx of the data. Round a and b to the nearest thousandth. [y = 20.274(1.037x)]
c. Use your model to estimate the year in which the whooping crane population reached 100 birds. [year 44, or 1984]
Part E, Making Connections
3. The California condor is another gravely endangered bird. Read the life history of the California condor.
a. Because of the success of captive breeding programs, the number of condors alive in 1987 soon doubled. How many years did it take the number of condors to double? [5 years]
b. Use the Rule of 70 to approximate the rate of increase in condor numbers during the doubling period. Assume continuous exponential growth. [14%]
c. Write and graph an equation that models continuous exponential growth during the population doubling period that began in 1987. [A = 27e 0.14t]
d. If your model is correct, how many condors will be alive in the year 2000? [167]
e. Is an exponential model a good one for modeling condor population growth? Explain. [Answers will vary. Probably not. Condors do not increase "continuously." Rather, condor females lay single eggs once or less per year (see "Life History"). Between hatchings, the population does not increase.]
In an effort to better understand the forces of the earth beneath us, seismologists--earthquake scientists--have gathered huge amounts of data on earthquakes.
Part B, Solving Exponential Equations
1. The Significant Earthquake Database contains information on more than 5000 destructive earthquakes that have occurred over the past 4000 years.
a. Search the database to find the Richter magnitudes of two quakes that took place in November, 1985--one in Turkey and one in the Vanuatu Islands. [Turkey, 4.1; Vanuatu Islands, 7.6]
b. The amplitude A of an earthquake shockwave is half the height of the wave. If R is the Richter magnitude of a quake and a is the amplitude of the normal background shockwave, R = log A / a. Let a = 1 unit.
Find the amplitudes of the Turkey and Vanuatu Islands earthquakes. Round to the nearest hundred. [Turkey, 12,600; Vanuatu Islands, 39,810,700]
c. Amplitude is a measure of the intensity of an earthquake. How many times stronger than the Turkey quake was the Vanuatu quake? [10 7.6 4.1 = 10 3.5 3162 times]
d. A quake approximately 200 times stronger than the Turkey quake took place in March, 1985. What was the Richter magnitude of the quake? In what group of islands did the quake take place? [6.4; Leeward Islands]
Part D, Making Connections
2a. The formula log10E = 11.4 + 1.5R approximates the amount of energy E in ergs, released in an earthquake of Richter magnitude R. Solve the formula for E. [E = 10 11.4 + 1.5R]
b. The largest earthquake in U.S. history took place in Alaska on March 28, 1964. Find the Richter magnitude of the quake. Approximate the amount of energy released by the quake. [9.2; E = 1.5849 X 1025 ergs]
c. Look at the bottom of the page of the Largest Earthquakes in the United States to find out which earthquake released 2.5119 X10 23 ergs of energy. Which earthquake was it, and what was its magnitude?
d. The velocity v, in mi/h, at which a tsunami (tidal wave) travels is related to the depth d in ft of the water through which the wave travels by the formula log14.9d v = 0.5. Find the velocity of the tsunami unleashed by the 1964 Alaska earthquake as the wave traveled across Pacific Ocean waters 3 miles in depth.
