Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Note Some of the formulas below use the special number
A Population of Deer When a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first, but to level out when the population grows to near the maximum that the environment can support. Such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. The number N of deer present at time 1 (measured in years since the herd was introduced) on a certain wildlife reserve has been determined by ecologists to be given by the function
Figure(1)
a. How many deer were initially on the reserve?
b. Calculate
c. Express the number of deer present after 15 years using functional notation, and then calculate it.
d. How much increase in the deer population do you expect from the
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to two decimal places unless otherwise indicated. Population Growth A population of animals is growing exponentially, and an ecologist has made the following table of the population size, in thousands, at the start of the given year. Year Population, in thousands 2011 5.25 2012 5.51 2013 5.79 2014 6.04 2015 6.38 2016 6.70 Looking over the table, the ecologist realizes that one of the entries for population size is in error. Which entry is it, and what is the correct population? Round the ratios to two decimal places.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Logistic Formula A population grows according to the logistic model. N=251+0.5e1.4t where t is measured in years and N is measured in thousands. a. What is r for this population? b. What is the environmental carrying capacity K? c. This population is subject to harvesting. What is the optimum yield level?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Nail Growth The rate of fingernail growth depends on many factors, but in adults, nails grow at an average rate of 3 millimeters per month. If a nail is initially 12 millimeters long, find a formula that gives the length L, in millimeters, of the nail if left unclipped after t months.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Continuous CompoundingThis is a continuation of Exercise 22. In this exercise, we examine the relationship between APR and the APY when interest is compounded continuously-in other words, at every instant. We will see by means of an example that the relationship is Yearlygrowthfactor=eAPR,(4.1) and so APY=eAPR1(4.2) if both the APR and the APY are in decimal form and interest is compounded continuously. Assume that the APR is 10, or 0.1 as a decimal. a.The yearly growth factor for continuous compounding is just the limiting value of the function given by the formula in part b of Exercise 22. Find that limiting value to four decimal places. b.Compute eAPR with an APR of 0.1 as a decimal. c.Use your answers to parts a and b to verify that Equation 4.1 holds in the case where the APR is 10. Note: On the basis of part a, one conclusion is that there is a limit to the increase in the yearly growth factor and hence in the APY as the number of compounding periods increases. We might have expected the APY to increase without limit for more and more frequent compounding. 22. APR and APYRecall that financial institutions sometimes report the annual interest rate that they offer on investments as the APR, often called the nominal interest rate. To indicate how an investment will actually grow, they advertise the annual percentage yield, or APY. In mathematical terms, this is the yearly percentage growth rate for the exponential function that models the account balance. In this exercise and the next, we study the relationship between the APR and the APY. We assume that the APR is 10. or 0.1 as a decimal. To determine the APY when we know the APR, we need to know how often interest is compounded. For example, suppose for the moment that interest is compounded twice a year. Then to say that the APR is 10 means that in half a year, the balance grows by 102 or 5. In other words, the 12-year percentage growth rate is 0.12 as a decimal. Thus, the 12-year growth factor is 1+0.12. To find the yearly growth factor, we need to perform a unit conversion: One year is 2 half-year periods, so the yearly growth factor is (1+0.12)2 or 1.1025. a.What is the yearly growth factor if interest is compounded four times a year? b.Assume that interest is compounded n times each year. Explain why the formula for the yearly growth factor is (1+0.1n)n. c.What is the yearly growth factor if interest is compounded daily? Give your answer to four decimal places.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Logistic Model A population grows according to the logistic model. The r value is 0.02 and the environmental carrying capacity is 2500. Write the logistic equation satisfied by the population if N(0)=100.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Gray Wolves in Idaho The report cited in Example 4.6 tells us that in 2009, there were 870 gray wolves in Idaho, but that the population declined by 19 that year. For purposes of this problem, we assume that this 19 annual rate of decrease continues. a. Find an exponential model that gives the wolf population W as function of the time t in years since 2009. b. It is expected that the wolf population cannot recover if there are fewer than 20 individuals. How long must this rate of decline continue for the wolf population to reach 20?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Minimum Monthly PaymentSuppose you have a balance of B dollars on credit card.You choose to stop charging and pay off the card, making only minimum monthly payments.If your card charges an APR of r, as a decimal, and requires a minimum monthly payment of 5 of the balance, then the time T, in months, required to reduce your balance to 100 is given by T=2logBlog(0.95(1+r/12)). Suppose your current balance is 8000. a.How long will it take to reduce your balance to 100 if the APR for your card is 25? Report your answer to the nearest whole month. b.Plot the graph of T versus r. Use a horizontal span of 0 to 0.3. c.Does a larger APR mean a longer or a shorter time to reduce the balance to 100?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_forwardReminder Round all answers to two decimal places unless otherwise indicated. The pH Scale Acidity of a solution is determined by the concentration H of hydrogen ions in the solution measured in moles per liter of solution. Chemists use the negative of the logarithm of the concentration of hydrogen ions to define the pH scale: pH=logH Lower pH values indicate a more acidic solution. a.Normal rain has a pH value of 5.6. Rain in the eastern United States often has a pH level of 3.8. How much more acidic is this than normal rain? b.If the pH of water in lake falls below a value of 5, fish often fail to reproduce. How much more acidic is this than normal water with a pH of 5.6?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. What if Interest is Compounded More Often than Monthly?Some lending institutions compound interest daily or even continuously. The term continuous compounding is used when interest is being added as often as possible-that is, at each instant in time. The point of this exercise is to show that, for most consumer loans, the answer you get with monthly compounding is very close to the right answer, even if the lending institution compounds more often. In part 1 of Example 1.2, we showed that if you borrow 7800 from an institution that compounds monthly at a monthly interest rate of 0.67 for an APR of 8.04 , then in order to pay off the note in 48months, you have to make a monthly payment of 190.57. a.Would you expect your monthly payment to be higher or lower if interest were compounded daily rather than monthly? Explain why. b.Which would you expect to result in a larger monthly payment, daily compounding or continuous compounding? Explain your reasoning. c.When interest is compounded continuously, you can calculate your monthly payment M=M(P,r,t) in dollars, for a loan of Pdollars to be paid off over t months using M=P(er1)1ert, where r=APR/12 if the APR is written in the decimal form. Use this formula to calculate the monthly payment on a loan of 7800 to be paid off over 48months with an APR of 8.04. How does this answer compare the result in Example 1.2?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. A Diet An overweight man makes lifestyle changes in order to lose weight. He currently weighs 260pounds, and he has set a target weight of 200pounds. Each month the difference D, in pounds, between his current weight and his target weight decreases by 10. a. Make an exponential model of D versus the time t in months since the diet began. b. How long will it take for his weight to reach 210poundsarrow_forwardReminder 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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning