Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
Traffic Flow For traffic moving along a highway, we use
a. An important measurement of the traffic on highway is the relative density
i. What does a value of
ii. What does a value of
b. Let
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
d. Traffic is considered to be seriously congested If the mean speed drops to
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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. 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. 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. Yellowfin Tuna Data were collected comparing the weight W, in pounds, of a yellowfin tuna to its length L, in centimeters. These data are presented in the following table. L=Length W=Weight 70 14.3 80 21.5 90 30.8 100 42.5 110 56.8 120 74.1 130 94.7 140 119 160 179 180 256 a. What is the average rate of change, in weight per centimeter of length, in going from a length of 100 centimeters to a length of 110 centimeters? b. What is the average rate of change, in weight per centimeter of length, in going from 160 to 180 centimeters? c. Judging from the data in the table, does an extra centimeter of length make more difference in weight for a small tuna or for a large tuna? d. Use the average rate of change to estimate the weight of a yellowtuna fish that is 167 centimeters long? e. What is the average rate of change, in length per pound of weight, in going from a weight of 179 pounds to a weight of 256 pounds? f. What would you expect to be the length of a yellow tuna weighing 225 pounds?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Arterial Blood Flow Medical evidence shows that a small change in the radius of an artery can indicate a large change in blood flow. For example, if one artery has a radius only 5 larger than another, the blood flow rate is 1.22 times as large. Further information is given in the table below. Increase in radius Times greater blood flow rate 5 1.22 10 1.46 15 1.75 20 2.07 a. Use the average rate of change to estimate how many times greater the blood flow rate is in an artery that has a radius 12 larger than another. b. Explain why if the radius is increased by 12 and then we increase the radius of the new artery by 12 again, the total increase in the radius is 25.44. c. Use parts a and b to answer the following question: How many times greater is the blood flow rate in an artery that 25.44 larger in radius than another? d. Answer the question in part c using the average rate of change.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Timber Values Under Scribner Scale The following table compares dollar values per standard cord (128cubicfeet) to values per thousand board-feet MBF under the Scribner scale for trees 12inches in diameter. Values per cord Values per MRF Scribner 20 68.00 24 81.60 28 95.20 36 122.40 a. Make a table showing the average rate of change for each interval of values. b. Would you expect the value per MBF Scribner to have a limiting value? c. If you are selling timber from trees 12inches in diameter, which is the better price: 25 per cord or 71 per MBF Scribner? What if you are buying the timber?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
- ReminderRound 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_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 answer to two decimal places unless otherwise indicated. Lean Body Weight in Males Your lean body weight L is the amount you would weigh if all the fat in your body were to disappear. One text gives the following estimate of lean body weight L in pounds for young adult males: L=98.42+1.08W4.14A, where W is total weight in pounds and A is abdominal circumference in inches. 7 a. Consider a group of young adult males who have the same abdominal circumference. If their weight increases but their abdominal circumference remains the same, how does their lean body weight change? b. Consider a group of young adult males who have the same weight. If their abdominal circumference decreases but their weight stays the same, how does their lean body weight change? c. Suppose a young adult male has a lean body weight of 144 pounds. Over a period of time, he gains 15 pounds in total weight, and his abdominal circumference increases by 2 inches. What is his lean body weight now?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Mosteller Formula for Body Surface Area Body surface area is an important piece of medical information because it is a factor in temperature regulation as well as some drug level determinations. The Mosteller formula gives one way of estimating body surface area B in square meters. The formula uses the weight w in kilograms and the height h in centimeters The relation is B=160h1/2w1/2 a. Use the Mosteller formula to estimate the body surface area for a man who is 188 centimeters tall and weighs 86kilograms. b. The weight of an adult increases by 10. How will this change affect his burly surface area? Give your answer as a percentage. Note: You may assume that the adults height will stay the same.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_forwardReminder Round all answers to two decimal places unless otherwise indicated. Estimating Wave Height Sailors use the following function to estimate wave height h, in feet, from wind speed w, in miles per hour h=0.02w2 a. Make a graph of wave height versus wind speed. Include wind speeds of up to 25 miles per hour. b. A small boat can sail safely provided wave heights are no more than 4 feet. What range of wind speed will give safe sailing for this boat?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning