Concept explainers
Reminder Round all the answers to two decimal places unless otherwise indicated.
Traffic Accidents The following table shows the rate
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a. Use regression to find a quadratic model for the data.
b. Calculate
c. At what speed is vehicular involvement in traffic accidents (for commercial vehicles driving at night on urban streets) at a minimum?
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Chapter 5 Solutions
FUNCTIONS+CHANGE -WEBASSIGN
- The American Food Dollar The following table shows the percentage P=P(d) of the American food dollar that was spent on eating away from home at restaurants, for example as function of the date d. d=Year P=Precentspentawayfromhome 1969 25 1989 30 2009 34 a. Find P(1989) and explain what it means. b. What does P(1999) mean? Estimate its value. c. What is the average rate of change per year in percentage of the food dollar spent away from home for the period from 1989 to 2009? d. What does P(2004) mean? Estimate its value. Hint: Your calculation in part c should be useful. e. Predict the value of P(2014) and explain how you made your estimate.arrow_forwardArctic Foxes The age-weight relationship of female Arctic foxes caught in Svalbard, Norway, can be estimated by the function Mt=3102e-e-0.022t-56, where t is the age of the fox in days and Mt is the weight of the fox in grams. Source: Journal of Mammalogy. a. Estimate the weight of a female fox that is 200 days old. b. Use Mt to estimate the largest size that a female fox can attain. Hint: Find limtMt. c. Estimate the age of a female fox when it has reached 80% of its maximum weight. d. Estimate the rate of change in weight of an Arctic fox that is 200 days old. Hint: Recall that Dteft=f'teft. e. Using a graphing calculator to graph Mt and then describe the growth pattern. f. Use the table function on a graphing calculator or a spreadsheet to develop a chart that shows the estimated weight and growth rate of female foxes for days 50,100,150,200,250, and 300.arrow_forwardMortgage Rates The following table is taken from the website of Freddie Mac. It shows rates for 30-year fixed-rate mortgages since 1970. y=Year r=Mortgagerate 1975 9.05 1980 13.74 1985 12.43 1990 10.13 1995 7.93 2000 8.05 2005 5.87 2010 4.69 2015 3.84 a. Explain in practical terms the meaning of r(2003). b. Use the table to estimate the value of r(2003).arrow_forward
- LIFE SCIENCE APPLICATIONS Brain Mass The brain mass of a fetus can be estimated using the total mass of the fetus by the function b=0.22m0.87, where m is the mass of the fetus in grams and b is the brain mass in grams. Suppose the brain mass of a 25-g fetus is changing at a rate of 0.25gperday. Use this to estimate the rate of change of the total mass of the fetus, dm/dt. Source: Archives dAnatomie, d Histologie et d Embryologie.arrow_forwardMass of Bighorn Yearlings The body mass of yearling bighorn sheep on Ram Mountain in Alberta, Canada, can be estimated by M(t)=27.5+0.3t0.001t2 Where M(t) is measured in kilograms and t is days since May 25. Source: Canadian Journal of zoology. a. Find the average rate of change of the mass of bighorn yearling between 105 and 115 days past May 25. b. Find the instantaneous rate of change of the mass for a bighorn yearling sheep whose age is 105 days past May 25. c. Graph the function M(t) on [5,125] by [25,65]. d. Does the behavior of the function past 125 days accurately model the mass of the sheep? Why or why not? e. f. g.arrow_forwardMolars The crown length as shown below of first molars in fetuses is related to the postconception age of tooth as L(t)=0.01t2+0.788t7.048, Where L(t) is the crown length, in millimeters, of the molar t weeks after conception. Source: American Journal of Physical Anthropology. a. Find the average rate of growth in crown length during weeks 22 through 28. b. Find the instantaneous rate of growth in crown length when the tooth is exactly 22 weeks of age. c. Graph the function on [0,50] by [0,9]. Does a function that increases and then begins to decrease make sense for this particular application? What do you suppose is happening during the first 11 weeks? Does this function accurately model crown length during those weeks?arrow_forward
- Farms in the United States The graph gives the number of farms in the United States from 1850 to 2000. aEstimate the average rate of change in the number of farms between i 1860 and 1890 and ii 1950 and 1970. bIn which decade did the number of farms experience the greatest average rate of decline?arrow_forwardThe Kelvin Temperature Scale Physicists and chemists often use the Kelvin temperature scale. In order to determine the relationship between the Fahrenheit and Kelvin temperature scales, a lab assistant put Fahrenheit and Kelvin thermometers side by side and took readings at various temperatures. The following data were recorded. K = kelvins F = degrees Fahrenheit 200 -99.67 220 -63.67 240 -27.67 260 8.33 280 44.33 300 80.33 a. Show that the temperature F in degrees Fahrenheit is a linear function of the temperature K in kelvins. b. What is the slope of this linear function? Note: Be sure to take into account that the table lists kelvins in jumps of 20 rather than in jumps of 1. c. Find a formula for the linear function. d. Normal body temperature is 98.6 degrees Fahrenheit. What is that temperature in kelvins? e. If temperature increases by 1 kelvin, by how many degrees Fahrenheit does it increase? If temperature increases by 1 degree Fahrenheit, by how many kelvins does it increase? f. The temperature of 0 kelvins is known as absolute zero. It is not quite accurate to say that all molecular motion ceases at absolute zero, but at that temperature the system has its minimum possible total energy. It is thought that absolute zero cannot be attained experimentally, although temperatures lower than 0.0000001 kelvin have been attained. Find the temperature of absolute zero in degrees Fahrenheit.arrow_forwardCortisone Concentration A mathematical model for the concentration of administered cortisone in humans over a 24 hour period uses the function C=DaV(ab)(ebteat) Where C is the concentration, D is the dose given at time t=0, V is the volume of distribution volume divided by bioavailability, a is the absorption rate, b is the elimination rate, and t is the time in hours. Source: Journal of Pharmacokinetics and Bio pharmaceutics. a. What is the value of C at t=0? Explain why this makes sense. b. What happens to the concentration as a large amount of time passes? Explain why this makes sense. c. The researchers used the values D=500micrograms,a=8.5,b=0.09 andV=3700liters. Use these values and a graphing calculator to estimate when the concentration is greatest.arrow_forward
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