Reminder Round all answers to two decimal places unless otherwise indicated.
Zipf’s Law The linguist George Kingsley Zipf (1902-1930) proposed a law that bears his name. In a typical English text, the most commonly occurring word is "the." We say that its frequency rank r is 1. The frequency
where
a. Use the frequency information given for "the" to determine the value of r
b. The third most common English word is "and." According to Zipf's law, what is the frequency of this word in a typical English text? Round your answer to one decimal place.
c. If one word's frequency rank is twice that of another, how do their frequencies of occurrence compare?
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Chapter 5 Solutions
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to decimal places unless otherwise indicated. The Spread of AIDS This table shows the cumulative number N=N(t) of AIDS cases in the United States that have been reported to the Centers for Disease Control and Prevention by the end of the year given. The source for these data, the U.S. Centers for Disease Control and Prevention in Atlanta, cautions that they are subject to retrospective change. a. What does dNdt mean in practical terms? b. From 2010to2014, was dNdt ever negative? t=year N=totalcasereported 2010 1,140,203 2011 1,172,489 2012 1,191,061 2013 1,217,863 2014 1,236,994arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Marriage Length In this exercise, we consider data from the Statistical Abstract of the United States on the fraction of women married for the first time in 1960 whose marriage reached a given anniversary number. The data show that the fraction of women who reached their fifth anniversary was 0.928. After that, for each one-year anniversary number, the fraction reaching that number drops by about 2. These data describe constant percentage change, so it is reasonable to model the fraction M as an exponential function of the number n of anniversaries since fifth. a.What is the yearly decay factor for the exponential model? b.Find an exponential model for M as a function of n. c.According to your model, what fraction of women married for the first time in 1960 celebrated their 40th anniversary? Take n=35. Round your answer to three decimal places. The actual fraction is 0.449 or 44.9.arrow_forwardReminderRound all answers to two decimal places unless otherwise indicated. Deaths from the Heart DiseaseTable A and B show the deaths per 100,000 caused by heart disease in the United States for males and females aged 55 to 64 years. The function Hm gives deaths per 100,000 for males, and Hf gives deaths per 100,000 for females. a.Approximate the value of dHmdt in 2004 using the average rate of change from 2004 to 2007. b.Explain the meaning of the number you calculated in part a in practical terms. You should, among other things, tell what the sign means. TABLE AHeart Disease Deaths per 100,000 for Males Aged 55 to 64 Years t=year Hm=deathsper100,000 1990 537.3 2000 371.7 2003 331.7 2004 312.8 2007 288.8 c.Use your answer from part a to estimate the heart disease death rate for males aged 55 to 64 years in 2006 d.Approximate the value of dHfdt for 2004 using the average rate of change from 2004 to 2007. e.Explain what your calculations from parts a and d tell you about comparing heart disease deaths for men and women in 2004. TABLE BHeart Disease Deaths per 100,000 for Females Aged 55 to 64 Years t=year Hf=deathsper100,000 1990 215.7 2000 159.3 2003 141.9 2004 131.5 2007 117.9arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Moores Law The speed of a computer chip is closely related to the number of transistors on the chip, and the number of transistors on a chip has increased with time in a remarkably consistent way. In fact, in the year 1965, Dr. Gordon E. Moore now chairman emeritus of Intel Corporation observed a trend and predicted that it would continue for a time. His observation, now known as Moores law, is that every two years or so a chip is introduced with double the number of transistors of its fastest predecessor. This law can be restated in the following way: If time increases by 1year, then the number of transistors is multiplied by 100.15.More generally, the rule is that if time increases by tyears, then the number of transistors is multiplied by 100.15t.For example, after 8years, the number of transistors is multiplied by 100.158, or about 16. The 6th generation Core processor was released by Intel Corporation in the year 2015. a.If a chip were introduced in the year 2022, how many times the transistors of the 6th generation Core would you expect it to have? Round your answer to the nearest whole number. b.The limit of conventional computing will be reached when the size of a transistors on a chip will be 200 times that of the 6th generation Core. When, according to Moores law, will that limit be reached? c.Even for unconventional computing, the law of physics impose a limit on the speed of computation. The fastest speed possible corresponds to having about 1040 times the number of transistors as on the 6th generation Core. Assume that Moores law will continue to be valid even for unconventional computing, and determine when this limit will be reached. Round your answer to the nearest century.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Defense SpendingData about recent federal defense spending are given in the accompanying Statistical Abstract of the United States table. Here t denotes the time, in years, since 1990 and D denotes federal defense spending, in billions of dollars. a.Calculate the average yearly rate of change in defense spending from 1990 to 1995. b.Use your answer from part a to estimate D(3), and explain what it means. t= Years since 1990 D= Spending billions of dollars 0 328.4 5 310.0 10 341.5 15 565.5 20 843.8 c.Calculate the average yearly rate of change in defense spending from 2005 to 2010. d.Use your answer form part c to estimate the value of D(22).arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. The Fukushima Disaster On March 11, 2011, Japan suffered an earthquake and tsunami that caused a disastrous accident at the Fukushima nuclear power plant. Among many other results, amounts of iodine-131 that were 27 times the government limit were found in a sample of spinach 60 miles away?' Now, 27 times the government limit of iodine-131 is 54 thousand becquerels per kilogram." The following table shows the amount I, in thousands of becquerels per kilogram, of iodine-131 that would remain after t days. t=time,indays I=amountofiodine-131 0 54.00 1 49.52 2 45.41 3 41.64 4 38.18 a. Show that the data are exponential. In this part and the next, round to three decimal places b. Find an exponential model that shows the amount of iodine-131 present after t days. c. How long will it take for the amount of iodine-131 to fall to the government limit of 2 thousand becquerels per kilogram? Round your answer to the nearest whole day.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. The half life of 239U Uranium-239 is an unstable isotope of uranium that decays rapidly. In order to determine the rate of decay, 1 gram of 239U was placed in a container, and the amount remaining was measured at 1-minute intervals and recorded in the table below. Time, in minutes Grams remaining 0 1 1 0.971 2 0.943 3 0.916 4 0.889 5 0.863 a. Show that these are exponential data and find an exponential model For this problem, round all your answers to three decimal places. b. What is the percentage decay rate each minute? What does this number mean in practical terms? c. Use functional notation to express the amount remaining after 10 minutes and then calculate the value. d. What is the half life of 239U?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Cleaning Contaminated Water A tank of water is contaminated with 60 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour, there is 22 less salt in the tank than at the beginning of the hour. Let S=S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins. a. Explain why S is an exponential function and find its hourly decay factor. b. Give a formula for S. c. Make a graph of S that shows the flushing process during the first 15 hours, and describe in words how the salt removal process progresses. d. In order to meet EPA standards, there can be no more than 3 pounds of salt in the tank. How long must the process continue before EPA standards are met? e. Suppose this cleanup procedure costs 8000 per hour to operate. How much does it cost to reduce the amount of salt from 60 pounds to 3 pounds? How much does it cost to reduce the amount of salt from 3 pounds to 0.1 pound?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Insect ControlDDT dichlorodiphenyltrichloroethane was used extensively from 1940 to 1970 as an insecticide. It still sees limited use for control of disease. But DDT was found to be harmful to plants and animals, including humans, and its effects were found to be lasting. The amount of time that DDT remains in the environment depends on many factors, but the following table shows what can be expected of 100 kilograms of DDT that has seeped into the soil. t=time,inyearssinceapplication D=DDTremaining,inkilograms 0 100.00 1 95.00 2 90.25 3 85.74 a. Show that the data are exponential. b. Make a model of D as an exponential function of t. c. What is the half-life of DDT in the soil? That is, how long will it be before only 50 kilograms of DDT remain?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Earthquakes in Alaska and ChileIn 1964, an earthquake measuring 9.2 on the Richter scale occurred in Alaska. a. How did the power of the New Madrid earthquake described in Example 4.12 compare with that of the 1964 Alaska earthquake? b. In 1960 an earthquake occurred in Chile that was 2 times as powerful as the Alaska quake. What was the Richter scale reading for the Chilean earthquake? EXAMPLE 4.12COMPARING SOME FAMOUS EARTHQUAKES On December 16, 1811, an earthquake occurred near New Madrid, Missouri, that temporarily reversed the course of the Mississippi River. This was actually one of a series of earthquakes in the area, one of which is estimated to have had a Richter magnitude of 8.8.The area was sparsely populated at the time, and there were thought to be few fatalities. On October 17, 1989, a calamitous earthquake measuring 7.1 on the Richter scale occurred in the San Francisco Bay area. The earthquake killed 67 and injured more than 3000. Part 1How much more powerful was the New Madrid quake than the 1989 San Francisco quake? Part 2If an earthquake 1000 times as powerful as the 1989 San Francisco earthquake occurred, what would its Richter scale measurement be? Part 3On December 26, 2004, an earthquake 2 times as powerful as the New Madrid quake struck the Indian Ocean near Indonesia. It caused a tsunami that resulted in the deaths of hundreds of thousands. What was the Richter scale reading for this quake?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Sales Growth A study of the sales s, in thousands of dollars, of a product as a function of time t, in years, yields the equation of change dsdt=0.3s(4s). This is valid for s less than 5. a.What level of sales will be attained in the long run? b.What is the largest rate of growth in sales?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Recent EarthquakesOn April 25, 2015, an earthquake of magnitude 7.8 on the Richter scale struck Nepal. On May 12, 2015, a major aftershock of magnitude 7.3 on the Richter scale shook the same region. a. How did the power of the first earthquake and this aftershock compare? b. What would be the magnitude of a quake twice as powerful as the first quake?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning