Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Note Some of the formulas below use the special number
Reynolds Number The Reynolds number is very important in such fields as fluid flow and aerodynamics. In the case of a fluid flowing through a pipe, the Reynolds’s number
Here
a. If the fluid in the pipe is toluene, its viscosity is
b. If the toluene is replaced by glycerol, then the viscosity is
Want to see the full answer?
Check out a sample textbook solutionChapter 1 Solutions
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to two decimal places unless otherwise indicated. Waiting at a Stop SignConsider a side road connecting to a major highway at a stop sign. According to a study by D.R. Drew, the average delay D, in seconds, for a car waiting at the stop sign to enter the highway is given by D=eqt1qtq, where q is the flow rate, or the number of cars per second passing the stop sign on the highway, and T is the critical highway that will allow for safe entry. We assume that the critical headway is T=5seconds. a.What is the average delay time if the flow rate is 500 cars per hour 0.14 car per second? b.The service rate s for a stop sign is the number of cars per second that can leave the stop sign. It is related to the delay by s=D1. Use function composition to represent the service rate as a function of flow rate. Reminder:(a/b)1=b/a. c.What flow rate will permit a stop sign service rate of 5 cars per minute 0.083 car per second?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. 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_forward
- Reminder 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_forwardReminder 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. 10.Flushing Chlorine City water, which is slightly chlorinated, is being used to flush a tank of heavily chlorinated water. The concentration C=C(t) of chlorine in the tank t hours after flushing begins is given by C=0.1+2.78e0.37tmilligramspergallon a. What is the initial concentration of chlorine in the tank? b. Express the concentration of chlorine in the tank after 3 hours using functional notation, and then calculate its value.arrow_forwardReminder Round all the answers to two decimal places unless otherwise indicated. Coxs Formula Assume that a long horizontal pipe connects the bottom of a reservoir with a drainage area. Coxs formula provides a way of determining the velocity v of the water flowing through the pipe: HdL=4v2+5v21200. Here H is the depth of the reservoir in feet, d is the pipe diameter in inches, L is the length of the pipe in feet, and the velocity v of the water is in feet per second. See Figure 5.71. a. Graph the quadratic function 4v2+5v2, using a horizontal span from 0 to 10. b. Judging on the basis of Coxs formula, is it possible to have a velocity of 0.25 foot per second? c. Find the velocity of the water in the pipe if the pipes diameter is 4 inches, its length is 1000 feet, and the reservoir is 50 feet deep. d. If the water velocity is too high, there will be erosion problems. Assuming that the pipe length is 1000 feet and the reservoir is 50 feet deep, determine the largest pipe diameter that will ensure that the water velocity does not exceed 10 feet per second.arrow_forward
- Reminder 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_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_forwardReminder 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. There is a formula that estimates how much your puppy will weigh when it reaches adulthood. The method we present applies to medium-sized breeds. First, find your puppys weight w, in pounds, at an age of a weeks, where a is 16 weeks or less. Then the predicted adult weight W=W(a,w) in pounds, is given by the formula. W=52wa a. Use functional notation to express the adult weight of a puppy that weighs 6 pounds at 14 weeks. b. Calculate the predicted adult weight for the puppy from part a.arrow_forward
- Reminder: Round all answer to two decimal places unless otherwise indicated. 3.Fat in Fast Food You are shopping for dinner for your family in a fast-food restaurant, but you are concerned about fat. An informative menu gives you the following information. Item Hamburger Chicken Sandwich Fries Onion rings Grams of fat 24 13 30 25 a. How much total fat is in an order of 3 hamburgers, 1 chicken sandwich, 2 orders of fries, and 2 orders of onion rings? b. Use a formula to express the total grams of fat F in an order of h hamburgers, c chicken sandwiches, f orders of fries, and o orders of onion rings.arrow_forwardReminder Round all answers to two decimals places unless otherwise indicated. Earths Umbra Earth has a shadow in space, just as people do on a sunny day. The darkest part 1 of that shadow is a conical region in space known as the umbra. A representation of Earths umbra is shown in Figure 3.24. Earth has radius of about 3960 miles, and the umbra ends at a point about 860,000 miles from Earth. The moon is about 239,000 miles from Earth and has a radius of about 1100 miles. Consider a point on the opposite side of Earth from the sun and at a distance from Earth equal to the moons distance from Earth. What is the radius of the umber at that point? Can the moon fit inside Earths umbra? What celestial event occurs when this happens?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. A Skydiver When a skydiver jumps from an airplane, his downward velocity increases until the force of gravity matches air resistance. The velocity at which this occurs is known as the terminal velocity. It is the upper limit on the velocity that a skydiver in free fall will attain in a stable, spread position, and tor a man 01 average size, its value is about 176 feet per second or 120 miles per hour. A skydiver jumped from an Airplane, and the difference D=D(t) between the terminal velocity and his downward velocity in feet per second was measured at 5-second intervals and recorded in the following table. t=seconds into free fall D=velocitydifference 0 176.00 5 73.61 10 30.78 15 12.87 20 5.38 25 2.25 a. Show that the data are exponential and find an exponential model for D. Round all your answers to two decimal places. b. W hat is the percentage decay rate per second for the velocity difference of the skydiver? Explain in practical terms what this number means. c. Let V=V(t) be the skydivers velocity t seconds into free fall. Find a formula for V. d. How long would it take the skydiver to reach 99 of terminal velocity?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning