Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Competition Between Populations In this exercise, we consider the problem of competition between two populations that vie for resources but do not prey on each other. Let
Per capita growth rate for
Per capita growth rate for
At an equilibrium point, the per capita growth rates for
Want to see the full answer?
Check out a sample textbook solutionChapter 3 Solutions
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to two decimal places unless otherwise indicated. The MacArthur-Wilson Theory of Biogeography Consider an island that is separated from the mainland, which contains a pool of potential colonizer species. The MacArthur-Wilson theory of biogeography hypothesizes that some species from the mainland will migrate to the island, but that increasing competition on the island will lead to species extinction. It further hypothesizes that both the rate of migration and the rate of extinction of species are exponential functions, and that an equilibrium occurs when the rate of extinction matches the rate of immigration. This equilibrium point is thought to be the point at which immigration and extinction stabilize. Suppose that, for a certain island near the mainland, the rate of immigration of new species is given by I=4.20.93tspeciesperyear and that the rate of species extinction on the island is given by E=1.51.1tspeciesperyear. According, to the MacArthur-Wilson theory, how long will be required for stabilization to occur, and what will be the immigration and extinction rates at that time?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. Profit The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. In this exercise, we measure all monetary values in dollars. To determine a formula for the total cost, we need to know the manufacturers fixed costs covering such things as plant maintenance and insurance as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. The total revenue R for a manufacturer during a given time period is also a function of the number N of items produced during that period. To determine a formula for the total monthly revenue, we need to know the selling price p per unit of the item, which in general is also a function of N. To find the total revenue, we multiply this selling price by the number of items produced. The profit P for a manufacturer is the total revenue minus the total cost. Suppose that a manufacturer of widgets has a fixed costs of 2000 per month and that the variable cost is 30 per widget. Further, the manufacturer has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. Number N Price p 200 41.00 250 40.50 300 40.00 350 39.50 a.Use a formula to express the total monthly cost C of this manufacturer as a function of N. b.Use the table to find a linear model of p as a function of N. c.Use your answer to part b to find a formula expressing the total monthly revenue R as function of N. d.Use your answers to part a and part c to find a formula expressing the monthly profit P as a function of N. What type of function is the profit: linear or quadratic? e.Find the two monthly production levels at which the manufacturer just breaks even that is, where the profit is zero.arrow_forwardReminderRound all answers to two decimal places unless otherwise indicated. NoteSome of the formulas below use the special number e, which was presented in the Prologue. Yield Response to Several Growth FactorsThis is a continuation of Exercise 23. If more than one nutrient is considered, the formula for percentage of maximum yield is a bit more complex. For three nutrients, the formula is Y(b,c,d)=(10.5b)(10.5c)(10.5d). a.Express using functional notation the percentage of maximum yield produced from 1 baule of the first nutrient, 2 baules of the third nutrient, and 3 baules of the third nutrient, and then calculate that value. b.One baule of nitrogen is 223 pounds per acre, 1 baule of phosphorus is 45 per acre, and 1 baule of potassium is 76 pounds per acre. What percentage of maximum yield will be obtained from 200 pounds of nitrogen per acre, 100 pounds of phosphorus per acre, and 150 pounds of potassium per acre? 23. Mitscherlichs EquationAn important agriculture problem is to determine how a quantity of nutrient, such as nitrogen, affects the growth of plants. We consider the situation wherein sufficient quantities of all but one nutrient are present. One boule of a nutrient is the amount needed to produce 50 of maximum possible yield. In 1990, E.A. Mitscherlichs proposed the following relation, which is known as Mitscherlichs Equation: Y=10.5b. Here b is the number of baules of nutrient applied, and Y is the percentage as a decimal of maximum yield produced. a.Verify that the formula predicts that 50 of maximum yield will be produced if 1 baule of nutrient is applied. b.Use functional notation to express the percentage of maximum yield produced by 3 baule of nutrient, and calculate the value. c.The exact value of a baule depends on the nutrient in question. For nitrogen, 1 baule is 223 pounds per acre. What percentage of maximum yield will be produced if 500 pounds of nitrogen per acre is present?arrow_forwardReminderRound all answers to two decimal places unless otherwise indicated. Protein Content of Wheat GrainProtein content of wheat grain is affected by soil moisture and the amount of available nitrogen among other things. Figure 1.45 shows" the percent of protein content of wheat grain versus pounds of nitrogen per acre applied in three separate situations. In each case, soil moisture refers to moisture at the soil depth of 2 inches to 12 inches. Situation 1: Irrigation was used when soil moisture dropped to 49. Situation 2: Irrigation was used when soil moisture dropped to 34. Situation 3: Irrigation was used when soil moisture dropped to 1. a. If irrigation begins when soil moisture reaches 49, what application of nitrogen will result in the lowest percentage of protein in wheat grain? b. If irrigation begins when soil moisture reaches 34, what application of nitrogen will result in the same protein content of wheat grain as beginning irrigation when soil moisture reaches 1? c. If you irrigate when soil moisture reaches 34, how much nitrogen should you apply to achieve a 13 protein content in wheat grain? d. Does Figure 1.45 indicate that, for nitrogen levels at 45 pounds per acre or higher, increased protein content in wheat grain is associated with higher or lower soil moisture? FIGURE 1.45 Protein content versus availability of nitrogenarrow_forward
- Reminder 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_forwardReminder Round all answers to two decimal places unless otherwise indicated. A Coin CollectionThe value of a coin collection increases as new coins are added and the value of some rare coins in the collection increases. The value V, in dollars, of the collection t years after the collection was started is given by the following table. t=time,inyears V=value,indollars 0 130.00 1 156.00 2 187.20 3 224.64 4 269.57 a. Show that these data are exponential. b. Find an exponential model for the data. c. According to the model, when will the collection have a value of 500?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Boron Uptake Many factors influence a plants uptake of boron from the soil, but one key factor is soil type. One experiment40 compared plant content C of boron, in parts per million, with the amount B, in parts per million, of water-soluble boron in the soil. In Decatur silty clay, the relation is given by C=33.78+37.5B. In Hartsells fine sandy loam, the relation is given by C=31.22+71.17B. a. What amount of available water-soluble boron will result in the same plant content of boron for Decatur silty clay and Hartsells fine sandy loam? If you choose to solve this problem graphically, we suggest a horizontal span of 0 to 0.5 for B b. For available boron amounts larger than that found in part a, which of the two soil types results in the larger plant content of boron?arrow_forward
- Reminder: Round all answer to two decimal places unless otherwise indicated. 15.Total Cost The total cost C for a manufacturer during a given time period is a function of the number N of items produced during that period. To deter mine a formula for the total cost, we need to know the manufacturers fixed costs covering things such as plant maintenance and insurance, as well as the cost for each unit produced, which is called the variable cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs. Suppose that a manufacturer of widgets has fixed costs of 9000 per month and that the variable cost is 15 per widget so it costs 15 to produce 1 widget. a. Use a formula to express the total cost C of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use. b. Express using functional notation the total cost if there are 250 widgets produced in a month, and then calculate that value.arrow_forwardReminder Round all answer to two decimal places unless otherwise indicated. Digitized Pictures on a Disk Drive The hard disk drive on a computer holds 800 gigabytes of information. That is 800,000 megabytes. The formatting information, operating system, and applications software take up 6000 megabytes of disk space. The operator wants to store on his computer a collection of digitized pictures, each of which requires 2 megabytes of storage space. a. We think of the total amount of storage space used on the disk drive as a function of the number of pictures that are stored on the drive. Explain why this function in linear. b. Find a formula to express the total amount of storage space used on the disk drive as a linear function of the number of pictures that are stored on the drive. Be sure to identify what the letters you use mean. Explain in practical terms what the slope of this function is. c. Express using functional notation the total amount storage space used on the disk drive if there are 350 pictures stored on the drive, and then calculate that value. d. After putting a number of pictures on the disk drive, the operator executes a directory command, and at the end of the list, the computer displays the message 769, 000, 000, 000 bytes free. This message means that there are 769,000 megabytes of storage space left on the computer. How many pictures are stored on the disk drive? How many additional pictures can be added before the disk drive is filled?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. World Copper Production World production of copper, in millions of tons per year, from 1900 to 2000 is given by C=0.51.033t, where t is the time in years since 1900. a.What production level does this model give for the year 2000? b.If this model were extended to 2025, how could you use your knowledge of copper production in 2024 to estimate copper production in 2025?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning