Reminder Round all answers to two decimal places unless otherwise indicated.
Waiting at a Stop Sign Consider a side road connecting to a major highway at a stop sign. According to a study by D.R. Drew, the average delay
where
a. What is the average delay time if the flow rate is
b. The service rate
Use function composition to represent the service rate as a function of flow rate. Reminder:
c. What flow rate will permit a stop sign service rate of
Want to see the full answer?
Check out a sample textbook solutionChapter 5 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 Dangers of Smoking Cigarette smoke contains any number of unhealthy substances, cyanide among them. One study modeled cyanide in the bloodstream after smoking a cigarette using C=0.1+0.3t0.6e0.17t, where C is the concentration of cyanide in the bloodstream, measured in nanograms per deciliter, and t is the time, in minutes, since smoking a cigarette. a. Make a graph of the concentration of cyanide during the first hour after smoking a cigarette. Add the line corresponding to the target level of 0.3 nanogram per deciliter. b. During which period is the concentration of cyanide 0.3 nanogram per deciliter or higher?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. Poiseuilles law for fluid velocitiesPoiseuilles law describes the velocities of fluids flowing in a tube---for example, the flow of blood in a vein. See Figure 5.74 This law applies when the velocities are not too large----more specifically, when the flow has no turbulence. In this case, the flow is laminar, which means that the paths of the flow are parallel to the tube walls. The law states that v=k(R2r2), where v is the velocity, k is a constant which depends on the fluid, the tube, and the units used for measurement, R is the radius of the tube, and r is the distance from the centerline of the tube. Since k and R are fixed for any application, v is a function at a point of distance r from the centerline of the tube. a.What is r for a point along the walls of the tube? What is the velocity of the fluid along the walls of the tube? b.Where in the tube does the fluid flow most rapidly? c.Choose numbers for k and R, and make a graph of v as a function of r. Use a horizontal span of 0 to R. d.Describe your graph from part c. e.Explain why you needed to use a horizontal span of 0 to R in order to describe the flow throughout the tube.arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Falling with a Parachute If an average-sized man jumps from an airplane with a properly opening parachute, his downward velocity v=v(t), in feet per second, t seconds into the fall is given by the following table. t=Secondsintothefall v=Velocity 0 0 1 16 2 19.2 3 19.84 4 19.97 a. Explain why you expect v to have a limiting value and what this limiting value represents physically. b. Estimate the terminal velocity of the parachutist.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Traffic FlowFor traffic moving along a highway, we use q to denote the mean flow rate. That is the average number of vehicles per hour passing a certain point. We let qm denote the maximum flow rate, k the mean traffic density that is, the average number of vehicles per mile, and km the density at which flow rate is a maximum that is, the value of k when q=qm. a.An important measurement of the traffic on highway is the relative density R, which is defined as R=kkm. i. What does a value of R1 indicate about traffic on highway? b.Let u denote the mean speed of vehicles on the road and uf the free speed-that is, the speed when there is no traffic congestion at all. One study proposes the following relation between density and speed: u=ufe0.5R2. Use function composition to find a formula that directly relates mean speed to mean traffic density. c.Make a graph of mean speed versus mean traffic density assuming that km is 122 cars per mile and uf is 75 miles per hour. include values mean traffic density up to 250 vehicles per mile Paying particular attention to concavity, explain the significance of the point k=122 on the graph. d.Traffic is considered to be seriously congested If the mean speed drops to 35 miles per hour. Use the graph from part c to determine what density will result in serious congestion.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Population Growth The following table shows the population of reindeer on an island as of the given year. date 1945 1950 1955 1960 population 40 165 678 2793 We let t be the number of years since 1945, so that t=0 corresponds to 1945, and we let N=N(t) denote the population size. a. Approximate dNdt for 1955 using the average rate of change from 1955 to 1960, and explain what this number means in practical terms. b. Use your work from part a to estimate the population in 1957. c. The number you calculated in part a is an approximation to the actual rate of change. As you will be asked to show in the next exercise, the reindeer population growth can be closely modeled by an exponential function. With this in mind, du you think your answer in part a is too large or too small? Explain your reasoning.arrow_forward
- ReminderRound 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_forwardReminder Round all answers to decimal places unless otherwise indicated. Mileage for an Old Car The gas mileage M that you get on your car depends on its age t in years. a. Explain the meaning of dMdt in practical terms. b. As your car ages and its performance degrades, do you expect dMdt to be positive or negative?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. The Photic Zone This is a continuation of Exercise 18. In the ocean, the photic zone is the region where there is sufficient light for photosynthesis to occur. See Figure 4.12. For marine phytoplankton, the photic zone extends from the surface of the ocean to a depth where the light intensity is about 1 of surface light. Near Cape Cod, Massachusetts, the depth of the photic zone is about 16meters. In waters near Cape Cod, by what percentage does light intensity decrease for each additional meter of depth?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_forwardReminder 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_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning