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Linda Almgren Kime, Judy Clarc, Beverly K. Michael, Explorations in College Algebra, 5th edition, Wiley 2011
11. Find an equation to represent the cost of attending college classes if application and registration fees are $150 and classes cost $250 per credit.
12. a. Write an equation that describes the total cost to produce x items if the startup cost is $200,000 and the production cost per item is $15.
b. Why is the total average cost per item less if the item is produced in large quantities?
13. Your bank charges you a $2.50 monthly maintenance fee on your checking account and an additional $0.10 for each check you cash. Write an equation to describe your monthly checking account costs.
14. If a town starts with a population of 63,500 that declines by 700 people each year, construct an equation to model its population size over time. How long would it take for the population to drop to 53,000?
15. A teacher’s union has negotiated a uniform salary increase for each year of service up to 20 years. If a teacher started at $26,000 and 4 years later had a salary of $32,000:
a. What was the annual increase?
b. What function would describe the teacher’s salary over time?
c. What would be the domain for the function?
16. Your favorite aunt put money in a savings account for you. The account earns simple interest; that is, it increases by a fixed amount each year. After 2 years your account has $8250 in it and after 5 years it has $9375. a. Construct an equation to model the amount of money in your account.
b. How much did your aunt put in initially?
c. How much will your account have after 10 years?
17. You read in the newspaper that the river is polluted with 285 parts per million (ppm) of a toxic substance, and local officials estimate they can reduce the pollution by 15 ppm each year.
a. Derive an equation that represents the amount of pollution, P, as a function of time, t.
b. The article states the river will not be safe for swimming until pollution is reduced to 40 ppm. If the cleanup proceeds as estimated, in how many years will it be safe to swim in the river?
18. The women’s recommended weight formula from Harvard Pilgrim Healthcare says, "Give yourself 100 lb for the first 5 ft plus 5 lb for every inch over 5 ft tall." a. Find a mathematical model for this relationship. Be sure you clearly identify your variables. b. Specify a reasonable domain and range for the function and then graph the function. c. Use your model to calculate the recommended weight for a woman 5 feet, 4 inches tall; and for one 5 feet, 8 inches tall.
19. In 1977 a math professor bought her condominium in Cambridge, Massachusetts, for $70,000. The value of the condo has risen steadily so that in 2007 real estate agents tell her the condo is now worth $850,000. a. Find a formula in point-slope form, to represent these facts about the value of the condo V(t), as a function of time, t, in years. b. If she retires in 2010, what does your formula predict her condo will be worth then?
1. Using the general formula y = mx that describes direct proportionality, find the value of m if:
a. y is directly proportional to x and y =2 when x = 10.
b. y is directly proportional to x and y = 0.1 when x = 0.2.
c. y is directly proportional to x and y = 1 when x = 1/4.
2. For each part, construct an equation and then use it to solve the problem. a. Pressure P is directly proportional to temperature T, and P is 20 lb per square inch when T is 60 degrees Kelvin. What is the pressure when the temperature is 80 degrees Kelvin? b. Earnings E are directly proportional to the time T worked, and E is $46 when T is 2 hours. How long has a person worked if she earned $471.50?
3. The number of centimeters of water depth W produced by melting snow is directly proportional to the number of centimeters of snow depth S. If W is 15.9 cm when S is 150 cm, then how many centimeters of water depth are produced by a 100-cm depth of melting snow?
4. The electrical resistance R (in ohms) of a wire is directly proportional to its length l (in feet). a. If 250 feet of wire has a resistance of 1.2 ohms, find the resistance for 150 ft of wire. b. Interpret the coefficient of l in this context.
5. For each of the following linear functions, determine the independent and dependent variables and then construct an equation for each function. a. Sales tax is 6.5% of the purchase price. b. The height of a tree is directly proportional to the amount of sunlight it receives. c. The average salary for full-time employees of American domestic industries has been growing at an annual rate of $1300/year since 1985, when the average salary was $25,000.
6. On the scale of a map 1 inch represents a distance of 35 miles. a. What is the distance between two places that are 4.5 inches apart on the map? b. Construct an equation that converts inches on the map to miles in the real world.
7. Find a function that represents the relationship between distance, d, and time, t, of a moving object using the data in the table at the top of the next column. Is d directly proportional to t? Which is a more likely choice for the object, a person jogging or a moving car?
9. Find the slope of the line through the pair of points, then determine the equation.
a. (2, 3) and (5, 3)
b. (–4, –7) and (12, –7)
c. (–3, 8) and (–3, 4)
d. (2, –3) and (2, –1)
12. An employee for an aeronautical corporation had a starting salary of $25,000/year. After working there for 10 years and not receiving any raises, he decides to seek employment elsewhere. Graph the employee’s salary as a function of time for the time he was employed with this corporation. What is the domain? What is the range?
13. For each of the given points write equations for three lines that all pass through the point such that one of the three lines is horizontal, one is vertical, and one has slope 2.
a. (1, –4)
b. (2, 0)
14. Consider the function f(x)= 4. a. What is f(0)? f(30)? f(–12.6)? b. Describe the graph of this function. c. Describe the slope of this function’s graph.
15. A football player who weighs 175 pounds is instructed at the end of spring training that he has to put on 30 pounds before reporting for fall training. a. If fall training begins 3 months later, at what (monthly) rate must he gain weight? b. Suppose that he eats a lot and takes several nutritional supplements to gain weight, but due to his metabolism he still weighs 175 pounds throughout the summer and at the beginning of fall training. Sketch a graph of his weight versus time for those 3 months.
16. a. Write an equation for the line parallel to y = 2 + 4x that passes through the point (3, 7). b. Find an equation for the line perpendicular to y = 2 + 4x that passes through the point (3, 7). 17.
a. Write an equation for the line parallel to y = 4 – x that passes through the point (3, 7).
b. Find an equation for the line perpendicular to y = 4 – x that passes through the point (3, 7).
18. Construct the equation of a line that goes through the origin and is parallel to the graph of the given equation.
a. y = 6
b. x = –3
c. y = –x + 3
19. Construct the equation of a line that goes through the origin and is perpendicular to the given equation.
a. y = 6
b. x = –3
c. y = –x + 3
20. Which lines are parallel to each other? Which lines are perpendicular to each other?
a. y = 1/3x + 2 c. y = –2x + 10 e. 2y + 4x = –12
b. y = 3x – 4 d. y = –3x – 2 f. y – 3x = 7
Two companies offer starting employees incentives to stay with the company after they are trained for their new jobs. Company A offers an initial hourly wage of $7.25, then increases the hourly wage by $0.15 per month. Company B offers an initial hourly wage of $7.70, then increases the hourly wage by $0.10 per month. a. Examine the accompanying graph. After how many months does it appear that the hourly wage will be the same for both companies? b. Estimate that hourly wage. c. Form two linear functions for the hourly wages in dollars of WA(m) for company A and WB(m) for company B after m months of employment. d. Does your estimated solution from part (a) satisfy both equations? If not, find the exact solution. e. What is the exact hourly wage when the two companies offer the same wage? f. Describe the circumstances under which you would rather work for company A and for company B.
2. Predict the number of solutions to each of the following systems. Give reasons for your answer. You don’t need to find any actual solutions.
a. y = 20,000 + 700x y = 15,000 + 800x
b. y = 20,000 + 700x y = 15,000 + 700x
c. y = 20,000 + 700x y = 20,000 + 800x
7.Assume you have $2000 to invest for 1 year. You can make a safe investment that yields 4% interest a year or a risky investment that yields 8% a year. If you want to combine safe and risky investments to make $100 a year, how much of the $2000 should you invest at the 4% interest? How much at the 8% interest?
8. Two investments in high-technology companies total $1000. If one investment earns 10% annual interest and the other earns 20%, find the amount of each investment if the total interest earned is $140 for the year.
10. For each of the following systems of equations, describe the graph of the system and determine if there is no solution, an infinite number of solutions, or exactly one solution.
a. 2x + 5y = –10 b. 3x + 4y = 5 c. 2x – y = 5
y = –0.4x – 2 3x – 2y = 5 6x – 3y = 4
11. If y = b + mx, solve for values of m and b by constructing two linear equations in m and b for the given sets of ordered pairs.
a. When x = 2, y = –2 and when x = –3, y = 13.
b. When x = 10, y = 38 and when x = 1.5, y = –4.5.
12. The following are formulas predicting future raises for four different groups of union employees. N represents the number of years from the start date of all the contracts. Each equation represents the salary that will be earned after N years.
Group A: Salary 5 49,000 1 1500N
Group B: Salary 5 49,000 1 1800N
Group C: Salary 5 43,000 1 1500N
Group D: Salary 5 37,000 1 2100N
a. Will group A ever earn more per year than group B? Explain.
b. Will group C ever catch up to group A? Explain.
c. How much total salary would an individual in each group have earned 3 years after the contract?
d. Will group D ever catch up to group C? If so, after how many years and at what salary?
e. Which group will be making the highest yearly salary in 5 years? How much will that salary be?
13. The supply and demand equations for a particular bicycle model relate price per bicycle, p (in dollars) and q, the number of units (in thousands). The two equations are
p = 250 + 40q Supply
p = 510 – 25q Demand
a. Sketch both equations on the same graph. On your graph identify the supply equation and the demand equation.
b. Find the equilibrium point and interpret its meaning.
14. For a certain model of DVD player, the following supply and demand equations relate price per player, p (in dollars) and number of players, q (in thousands).
p = 50 + 2q Supply
p = 155 – 5q Demand
a. Find the point of equilibrium.
b. Interpret this result.
17. A restaurant is located on ground that slopes up 1 foot for every 20 horizontal feet. The restaurant is required to build a wheelchair ramp starting from an entry platform that is 3 feet above ground. Current regulations require a wheelchair ramp to rise up 1 foot for every 12 horizontal feet, (See accompanying figure where H = height in feet, d = distance from entry in feet, and the origin is where the H-axis meets the ground.) Where will the new ramp intersect the ground?
22. Nenuphar wants to invest a total of $30,000 into two savings accounts, one paying 6% per year in interest and the other paying 9% per year in interest (a more risky investment). If after 1 year she wants the total interest from both accounts to be $2100, how much should she invest in each account?
23. When will the following system of equations have no solution? Justify your answer.
y=m1x + b1
y=m2x + b2
24. a. Construct a system of linear equations where both of the following conditions are met: The coordinates of the point of intersection are (2, 5). One of the lines has a slope of –4 and the other line has a slope of 3.5.
b. Graph the system of equations you found in part (a). Verify that the coordinates of the point of intersection are the same as the coordinates specified in part (a).
23. A financial advisor has up to $30,000 to invest, with the stipulation that at least $5000 is used to purchase Treasury bonds and at most $15,000 in corporate bonds.
a. Construct a set of inequalities that describes the relationship between buying corporate vs. Treasury bonds, where the total amount invested must be less than or equal to $30,000. (Let C be the amount of money invested in corporate bonds, and T the amount invested in Treasury bonds.).
b. Construct a feasible region of investment; that is, shade in the area on a graph that satisfies the spending constraints on both corporate and Treasury bonds. Label the horizontal axis "Amount invested in Treasury bonds" and the vertical axis "Amount invested in corporate bonds."
c. Find all of the intersection points (corner points) of the bounded investment feasibility region and interpret their meanings.
24. A Texas oil supplier ships at most 10,000 barrels of oil per week. Two distributors need oil. Southern Oil needs at least 2000 barrels of oil per week and Regional Oil needs at least 5000 barrels of oil per week. a. Let S be the number of barrels of oil shipped to Southern Oil and let R be the number of barrels shipped to Regional Oil per week. Create a system of inequalities that describes all of the conditions. b. Graph the feasible region of the system. c. Choose a point inside the region and describe its meaning.
25. A small T-shirt company created the following cost and revenue equations for a line of T-shirts, where cost C is in dollars for producing x units and revenue R is in dollars from selling x units:
C = 512.5x + 360 and R =15.5x
a. What does 12.5 represent?
b. What does 15.5 represent?
c. Find the breakeven point.
d. What is the cost of producing x units at the breakeven point? The revenue at the breakeven point?
e. Graph C and R on the same grid and shade the region that represents profit.
26. A large wholesale nursery sells shrubs to retail stores. The cost C(x) and revenue R(x) equations (in dollars) for x shrubs are
C(x)=15x + 12,000 and R(x)=18x
a. Find the breakeven point. b. Explain the meaning of the coordinates for the breakeven point. c. Graph C(x) and R(x) on the same grid and shade the region that represents loss.
27. An equilateral triangle has sides of length 8 cm. a. Find the height of the triangle. b. Find the area A of the triangle if A = (1/2)bh
28. An Egyptian pyramid consists of a square base and four triangular sides. A model of a pyramid is constructed using four equilateral triangles each with a side length of 30 inches. Find the surface area of the pyramid model, including the base.
8. $5,000 is invested in an institution that promises to double the investment every 8 years.
a. Assuming no withdrawals, find the balance after 8, 16, and 24 years.
b. Write a function, f(T), that gives the balance using T to represent 8-year periods.
c. Use this function to determine the balance after 40 years.
9. A tuberculosis culture increases by a factor of 1.185 each hour. a. If the initial concentration is 5 × 103 cells/ml, construct an exponential function to describe its growth over time. b. What will the concentration be after 8 hours?
10. An ancient king of Persia was said to have been so grateful to one of his subjects that he allowed the subject to select his own reward. The clever subject asked for a grain of rice on the first square of a chessboard, two grains on the second square, four on the next, and so on.
a. Construct a function that describes the number of grains of rice, G, as a function of the square, n, on the chessboard. (Note: There are 64 squares.)
b. Construct a table recording the numbers of grains of rice on the first ten squares.
c. Sketch your function.
d. How many grains of rice would the king have to provide for the 64th (and last) square?
11. Suppose you accumulate $2000 in debt on your credit card at an interest rate of 3.25% compounded monthly. If you are not able to pay off this debt, the amount of money, A, owed to the credit card company after m months is A = 2,000 ´ 1.0325m. Graph A and use the graph to explain why it is unwise to accumulate credit card debt without paying off some of the debt each month. In approximately how many months would the interest accumulated be equal to the original amount of your debt?
10. The per capita (per person) consumption of milk was 28 gallons in 1980 and has been steadily decreasing by an annual decay factor of 0.99.
a. Form an exponential function for per capita milk consumption M(t) for year t after 1980.
b. According to your function, what was the per capita consumption of milk in 2010? If available, use the Internet to check your predictions.
11. A 2.5-gram sample of an isotope of strontium-90 was formed in a 1960 explosion of an atomic bomb at Johnson Island in the Pacific Test Site. The half-life of strontium-90 is 28 years. In what year will only 0.625 gram of this strontium-90 remain?
15. Given the following table of values, where t = number of minutes and Q(t) = amount of substance remaining in grams after t minutes:
a. What is the initial value of Q(t)?
b. What is the half-life?
c. Construct an exponential decay function Q(t), where t is measured in minutes.
d. What is Q(5)? What does it represent?
16. The half-life of francium is 21 minutes. Starting with 4 × 1018 atoms of francium, how many atoms would disintegrate in 1 hour and 45 minutes? What fraction of the original sample remains?
18. It takes 1.31 billion years for radioactive potassium-40 to drop to half its original size.
a. Construct a function to describe the decay of potassium-40.
b. Approximately what amount of the original potassium-40 would be left after 4 billion years? Justify your answer.
4. A herd of deer has an initial population of 10 at time t = 0, with t in years.
a. If the size of the herd increases by 8 per year, find the formula for the population of deer, P(t), over time.
b. If the size of the herd increases by a factor of 1.8 each year, find the formula for the deer population, Q(t), over time.
c. For each model create a table of values for the deer population for a 10-year period.
d. Using the table, estimate when the two models predict the same population size.
5. Construct both a linear and an exponential function that go through the points (0, 6) and (1, 9).
6. Construct both a linear and an exponential function that go through the points (0, 200) and (10, 500).
7. Find the equation of the linear function and the exponential function that are sketched through two points on each of the following graphs.
13. Mute swans were imported from Europe in the nineteenth century to grace ponds. Now there is concern that their population is growing too rapidly, edging out native species. During the 2000s, many states created mute swan controls. The goal was to reduce the mute swan count from 14,313 in 2002 to 3000 swans by 2013.
a. Using a linear model, what would be the average rate of change per year needed to bring the swan population to that level? b. Using an exponential model, what would be the decay factor needed?
14. The price of a home in Carnegie, PA was $60,000 in 1980 and rose to $120,000 in 2010.
a. Create two models, f(t) assuming linear growth and g(t) assuming exponential growth, where t = number of years after 1980.
b. Fill in the following table representing linear growth and exponential growth for t years after 1980.
17. A pollutant was dumped into a lake, and each year its amount in the lake is reduced by 25%.
a. Construct a general formula to describe the amount of pollutant after n years if the original amount is Ao.
b. How long will it take before the pollution is reduced to below 1% of its original level? Justify your answer.
18. A swimming pool is initially shocked with chlorine to bring the chlorine concentration to 3 ppm (parts per million). Chlorine dissipates in reaction to bacteria and sun at a rate of about 15% per day. Above a chlorine concentration of 2 ppm, swimmers experience burning eyes, and below a concentration of 1 ppm, bacteria and algae start to proliferate in the pool environment.
a. Construct an exponential decay function that describes the chlorine concentration (in parts per million) over time.
b. Construct a table of values that corresponds to monitoring chlorine concentration for at least a 2-week period.
c. How many days will it take for the chlorine to reach a level tolerable for swimmers? How many days before bacteria and algae will start to grow and you will need to add more chlorine? Justify your answers.
19. a. If the inflation rate is 0.7% a month, what is it per year? b. If the inflation rate is 5% a year, what is it per month?
22. The exponential function Q(T)=600(1.35)T represents the growth of a species of fish in a lake, where T is measured in 5-year intervals.
a. Determine Q(1), Q(2), and Q(3).
b. Find another function q(t), where t is measured in years.
c. Determine q(5), q(10), and q(15).
d. Compare your answers in parts (a) and (c) and describe your results.
1. Construct a function that would represent the resulting value:
a. If you invested $5000 for n years at an annually compounded interest rate of:
b. If you make three different $5000 investments today at the three different interest rates listed in part (a), how much will each investment be worth in 40 years?
2. A bank compounds interest annually at 4%
a. Write an equation for the value V of $100 in t years.
b. Write an equation for the value V of $1000 in t years.
c. After 20 years will the total interest earned on $1000 be ten times the total interest earned on $100? Why or why not?
5. You are looking for a safe place to put $20,000.00 for one year. You go online and find that three banks are offering the following rates for CDs. Find the effective rate for each to determine which would earn you the most interest at the end of one year.
Bank A 2.46% continuously
Bank B 2.48% quarterly
Bank C 2.47% monthly
6. In 2010 the Federal Reserve was keeping interest rates low to stimulate the economy. As a result banks were posting very low interest rates on certificate of deposits (CD).
a. By law banks must state both the nominal rate and the effective (APY) rates. On August 3, 2010 the following banks listed their CD rates as follows: Is the APY correct? Calculate the APY rates for each bank and explain why they were giving the same nominal and APY rates, since APY rates are generally higher.
b. What would be the APY for a bank whose rate was 0.60% compounded continuously?
c. Go online or check at your bank to find the current CD rates for banks in your area. Check the APY values to make sure the bank is posting it correctly.
9. Assume $10,000 is invested at a nominal interest rate of 8.5%. Write the equations that give the value of the money after n years and determine the effective interest rate if the interest is compounded:
10. Assume you invest $2000 at 3.5% compounded continuously.
a. Construct an equation that describes the value of your investment at year t.
b. How much will $2000 be worth after 1 year? 5 years? 10 years?
11. Construct functions for parts (a) and (b) and compare them in parts (c) and (d).
a. $25,000 is invested at 5.75% compounded quarterly.
b. $25,000 is invested at 5.75% compounded continuously.
c. What is the amount in each account at the end of 5 years?
d. Explain, using the concept of effective rates, why one amount is larger than the other.
14. Assume that $5000 was put in each of two accounts. Account A gives 4% interest compounded semiannually. Account B gives 4% compounded continuously.
a. What are the total amounts in each of the accounts after 10 years?
b. Show that account B gives 0.04% more interest annually than account A.
16. You want to invest money for your newborn child so that she will have $50,000 for college on her 18th birthday. Determine how much you should invest if the best annual rate that you can get on a secure investment is:
a. 6.5% compounded annually
b. 9% compounded quarterly
c. 7.9% compounded continuously
17. A city of population 1.5 million is expected to experience a 15% decrease in population every 10 years.
a. What is the 10-year decay factor? What is the yearly decay factor? The yearly decay rate?
b. Use part (a) to create an exponential population model g(t) that gives the population (in millions) after t years. c. Create an exponential population model h(t) that gives the population (in millions) after t years, assuming a 1.625% continuous yearly decrease.
d. Compare the populations predicted by the two functions after 20 years. What can you conclude?
24. The yearly per capita consumption of whole milk in the United States reached a peak of 40 gallons in 1945, at the end of World War II. By 1970 consumption was only 27.4 gallons per person. It has been steadily decreasing since 1970 at a rate of about 3.9% per year.
a. Construct an exponential model M(t) for per capita whole milk consumption (in gallons) where t = years since 1970.
b. Use your model to estimate the year in which per capita whole milk consumption dropped to 6 gallons per person. How does this compare with the actual consumption of 6.1 gallons per person in 2007?
c. What might have caused this decline?
27. In Chapter 5 we saw that the function N =No ×10T described the actual number N of E. coli bacteria in an experiment after T time periods (of 20 minutes each) starting with an initial bacteria count of No.
a. What is the doubling time?
b. How long would it take for there to be ten times the original number of bacteria?
28. A woman starts a training program for a marathon. She starts in the first week by doing 10-mile runs. Each week she increases her run length by 20% of the distance for the previous week.
a. Write a formula for her run distance, D, as a function of week, W.
b. Use technology to graph your function, and then use the graph to estimate the week in which she will reach a marathon length of approximately 26 miles.
c. Now use your formula to calculate the week in which she will start running 26 miles.
29. The half-life of bismuth-214 is about 20 minutes.
a. Construct a function to model the decay of bismuth-214 over time. Be sure to specify your variables and their units.
b. For any given sample of bismuth-214, how much is left after 1 hour?
c. How long will it take to reduce the sample to 25% of its original size?
d. How long will it take to reduce the sample to 10% of its original size?
30. A department store has a discount basement where the policy is to reduce the selling price, S, of an item by 10% of its current price each week. If the item has not sold after the tenth reduction, the store gives the item to charity.
a. For a $300 suit, construct a function for the selling price, S, as a function of week, W.
b. After how many weeks might the suit first be sold for less than $150? What is the selling price at which the suit might be given to charity?
31. If you drop a rubber ball on a hard, level surface, it will usually bounce repeatedly. (See the accompanying graph below.) Each time it bounces, it rebounds to a height that is a percentage of the previous height. This percentage is called the rebound height.
a. Assume you drop the ball from a height of 5 feet and that the rebound height is 60%. Construct a table of values that shows the rebound height for the first four bounces.
b. Construct a function to model the ball’s rebound height, H, on the nth bounce.
c. How many bounces would it take for the ball’s rebound height to be 1 foot or less?
d. Construct a general function that would model a ball’s rebound height H on the nth bounce, where Ho is the initial height of the ball and r is the ball’s rebound height (in decimal form).
17. How long would it take $15,000 to grow to $100,000 if invested at 8.5% compounded continuously?
18. The effective annual interest rate on an account compounded continuously is 4.45%. Estimate the nominal interest rate.
19. The effective annual interest rate on an account compounded continuously is 3.38%. Estimate the nominal interest rate.
20. A town of 10,000 grew to 15,000 in 5 years. Assuming exponential growth:
a. What is the annual growth rate? b. What was its annual continuous growth rate?
29. The barometric pressure, p, in millimeters of mercury, at height h, in kilometers above sea level, is given by the equation p = 760e–0.128h. At what height is the barometric pressure 200 mm? 30. After t days, the amount of thorium-234 in a sample is A(t) = 35e–0.029t micrograms.
a. How much was there initially?
b. How much is there after a week?
c. When is there just 1 microgram left?
d. What is the half-life of thorium-234?
8. The stellar magnitude M of a star is approximately –2.5 log(B/B0), where B is the brightness of the star and B0 is a constant.
a. If you plotted B on the horizontal and M on the vertical axis, where would the graph cross the B axis?
b. Without calculating any other coordinates, draw a rough sketch of the graph of M. What is the domain?
c. As the brightness B increases, does the magnitude M increase or decrease? Is a sixth-magnitude star brighter or dimmer than a first-magnitude star?
d. If the brightness of a star is increased by a factor of 5, by how much does the magnitude increase or decrease?
9. If you listen to a 120-decibel sound for about 10 minutes, your threshold of hearing will typically shift from 0 dB up to 28 dB for a while. If you are exposed to a 92-dB sound for 10 years, your threshold of hearing will be permanently shifted to 28 dB. What intensities correspond to 28 dB and 92 dB?
12. An ulcer patient has been told to avoid acidic foods. If he drinks coffee, with a pH of 5.0, it bothers him, but he can tolerate both tap water, with a pH of 5.8, and milk, with a pH of 6.9.
a. Will a mixture of half coffee and half milk be at least as tolerable as tap water?
b. What pH will the half coffee–half milk mixture have?
c. In order to make 10 oz of a milk-coffee drink with a pH of 5.8, how many ounces of each are required?
13. Lemon juice has a pH of 2.1. If you make diet lemonade by mixing 1/4 cup of lemon juice with 2 cups of tap water, with a pH of 5.8, will the resulting acidity be more or less than that of orange juice, with a pH of 3?
13. The cost, C, in dollars, of insulating a wall is directly proportional to the area, A, of the wall (measured in ft2 ) and the thickness, t, of the insulation (measured in inches).
a. Write a cost equation for insulating a wall.
b. If the insulation costs are $12 when the area is 50 ft2 and the thickness is 4 inches, find the constant of proportionality.
c. A storage room is 15 ft by 20 ft with 8-ft-high ceilings. Assuming no windows and one uninsulated door that is 3 feet by 7 feet, what is the total area of the insulated walls for this room?
d. What is the cost of insulating this room with 4-inch insulation? With 6-inch insulation?
8. The time, t, required to empty a tank is inversely proportional to r, the rate of pumping. If a pump can empty the tank in 30 minutes at a pumping rate of 50 gallons per minute, how long will it take to empty the tank if the pumping rate is doubled?
9. The intensity of light from a point source is inversely proportional to the square of the distance from the light source. If the intensity is 4 watts per square meter at a distance of 6 m from the source, find the intensity at a distance of 8 m from the source. Find the intensity at a distance of 100 m from the source.
11. A light fixture is mounted flush on a 10-foot-high ceiling over a 3-foot-high counter. How much will the illumination (the light intensity) increase if the light fixture is lowered to 4 feet above the counter?
12. The frequency, f (the number of oscillations per unit of time), of an object of mass m attached to a spring is inversely proportional to the square root of m.
a. Write an equation describing the relationship.
b. If a mass of 0.25 kg attached to a spring makes three oscillations per second, find the constant of proportionality.
c. Find the number of oscillations per second made by a mass of 0.01 kg that is attached to the spring discussed in part (b).
13. Boyle’s Law says that if the temperature is held constant, then the volume, V, of a fixed quantity of gas is inversely proportional to the pressure, P. That is, for some constant k. What happens to the volume if:
a. The pressure triples?
b. The pressure is multiplied by n?
c. The pressure is halved?
d. The pressure is divided by n?
14. The pressure of the atmosphere around us is relatively constant at 15 lb/in2 at sea level, or 1 atmosphere of pressure (1 atm). In other words, the column of air above 1 square inch of Earth’s surface is exerting 15 pounds of force on that square inch of Earth. Water is considerably more dense. As we saw in Section 7.5, pressure increases at a rate of 1 atm for each additional 33 feet of water. The accompanying table shows a few corresponding values for water depth and pressure.
a. What type of relationship does the table describe?
b. Construct an equation that describes pressure, P, as a function of depth, D. c. In Section 7.5 we looked at a special case of Boyle’s Law for the behavior of gases, P=1/V (where V is in cubic feet, P is in atms).
15. a. Construct an equation to represent a relationship where x is directly proportional to y and inversely proportional to z.
b. Assume that x = 4 when y = 16 and z = 32. Find k, the constant of proportionality.
c. Using your equation from part (b), find x when y = 25 and z = 5.
16. a. Construct an equation to represent a relationship where w is directly proportional to both y and z and inversely proportional to the square of x.
b. Assume that w = 10 when y = 12, z = 15, and x = 6. Find k, the constant of proportionality.
c. Using your equation from part (b), find x when w = 2, y = 5, and z = 6.
20. The weight of a body is inversely proportional to the square of the distance from the body to the center of Earth. Assuming an Earth radius of 4000 miles, a man who weighs 200 pounds on Earth’s surface would weigh how much 20 miles above Earth?
12. A parabolic reflector 3" in diameter and 2" deep is proposed for a spotlight.
a. What formula is needed for manufacturing the reflector?
b. Where would the focus be?
c. Will a 1/4"-diameter light source fit if it is centered at the focus?
13. A slimline fluorescent bulb 1/2" in diameter needs 1" clearance top and bottom in a parabolic reflecting shade. a. What are the coordinates of the focus for this parabola? b. What is the equation for the parabolic curve of the reflector? c. What is the diameter of the opening of the shade?
20. A gardener wants to grow carrots along the side of her house. To protect the carrots from wild rabbits, the plot must be enclosed by a wire fence. The gardener wants to use 16 feet of fence material left over from a previous project. Assuming that she constructs a rectangular plot, using the side of her house as one edge, estimate the area of the largest plot she can construct.
21. Which of the following are true statements for quadratic functions?
a. The vertex and focal point always lie on the axis of symmetry.
b. The graph of a parabola could have three horizontal intercepts.
c. The graph of a parabola does not necessarily have a vertical intercept.
d. If f(2)= 0, then f has a horizontal intercept at 2.
e. The focal point always lies above the vertex.
22. The management of a company is negotiating with a union over salary increases for the company’s employees for the next 5 years. One plan under consideration gives each worker a raise of $1500 per year. The company currently employs 1025 workers and pays them an average salary of $30,000 a year. It also plans to increase its workforce by 20 workers a year.
a. Construct a function C(t) that models the projected cost of this plan (in dollars) as a function of time t (in years).
b. What will the annual cost be in 5 years?
5. Identify the stretch/compression factor and the vertex for each of the following:
a. y1 = 0.3(x – 1)2 c. y3 = 0.01(x + 20)2
b. y2 = 30x2 – 11 d. y2 = –6x2 + 12x
7. For each of the following quadratic functions, find the vertex (h, k) and determine if it represents the maximum or
minimum of the function.
a. ƒ(x) = –2(x – 3)2 c. ƒ (x) = –5(x + 4)2 – 7
c. ƒ(x) = 1.6(x +1)2 + 8 d. ƒ (x) = 8(x – 2)2 – 6
13. a. Find the equation of the parabola with a vertex of (2, 4) that passes through the point (1, 7).
b. Construct two different quadratic functions both with a vertex at (2, – 3) such that the graph of one function is concave up and the graph of the other function is concave down.
14. For each part construct a function that satisfies the given conditions.
a. Has a constant rate of increase of $15,000/year.
b. Is a quadratic that opens upward and has a vertex at (1, –4).
c. Is a quadratic that opens downward and the vertex is on the x-axis.
d. Is a quadratic with a minimum at the point (10, 50) and a stretch factor of 3.
e. Is a quadratic with a vertical intercept of (0, 3) that is also the vertex.
15. If a parabola is the graph of the equation y = a(x – 4)2 – 5:
a. What are the coordinates of the vertex? Will the vertex change if a changes?
b. What is the value of stretch factor a if the y-intercept is (0, 3)?
c. What is the value of stretch factor a if the graph goes through the point (1, –23)?
17. Determine the equation of the parabola whose vertex is at (2, 3) and that passes through the point (4, –1). Show your work, including a sketch of the parabola.
18. Students noticed that the path of water from a water fountain seemed to form a parabolic arc. They set a flat surface at the level of the water spout and measured the maximum height of the water from the flat surface as 8 inches and the distance from the spout to where the water hit the flat surface as 10 inches. Construct a function model for the stream of water.
1. Convert the following quadratic functions to vertex form.
Identify the coordinates of the vertex.
a. y =x2 + 8x + 11 b. y = 3x2 + 4x – 2
2. The daily profit, f (in dollars), of a hot pretzel stand is a function of the price per pretzel, p (in dollars), given by ƒ (p) = –1875p2 + 4500p – 2400.
a. Find the coordinates of the vertex of the parabola.
b. Give the maximum profit and the price per pretzel that gives that profit.
3. Find the equations in vertex form of the parabolas that satisfy the following conditions. Then check your solutions by using a graphing program, if available.
a. The vertex is at (–1, 4) and the parabola passes through the point (0, 2).
b. The vertex is at (1, –3) and one of the parabola’s two horizontal intercepts is at (–2, 0).
10. Find two different equations for a parabola that passes through the points (–2, 5) and (4, 5) and that opens downward.
14. Tom has a taste for adventure. He decides that he wants to bungee-jump off the Missouri River bridge. At time t (in seconds from the moment he jumps) his height h(t) (in feet above the water level) is given by the function h(t) = 20.5t2 – 123t + 190.5. How close to the water will Tom get?
15. A manager has determined that the revenue R(x) (in millions of dollars) made on the sale of supercomputers is given by R(x) = 48x – 3x2, where x represents the number of supercomputers sold. How many supercomputers must be sold to maximize revenue? According to this model, what is the maximum revenue (in millions of dollars)?
16. A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle (see the accompanying figure at the top of next column). If the perimeter of the window is 20 feet (including the semicircle), what dimensions will admit the most light (maximize the area)?
17. A pilot has crashed in the Sahara Desert. She still has her maps and knows her position, but her radio is destroyed. Her only hope for rescue is to hike out to a highway that passes near her position. She needs to determine the closest point on the highway and how far away it is.
a. The highway is a straight line passing through a point 15 miles due north of her and another point 20 miles due east. Draw a sketch of the situation on graph paper, placing the pilot at the origin and labeling the two points on the highway.
b. Construct an equation that represents the highway (using x for miles east and y for miles north).
c. Now use the Pythagorean Theorem to describe the square of the distance, d, of the pilot to any point (x, y) on the highway.
d. Substitute the expression for y from part (b) into the equation from part (c) in order to write d2 as a quadratic in x.
e. If we minimize d2 , we minimize the distance d. So let D = d2 and write D as a quadratic function in x. Now find the minimum value for D.
f. What are the coordinates of the closest point on the highway, and what is the distance, d, to that point?
28. Market research suggests that if a particular item is priced at x dollars, then the weekly profit P(x), in thousands of dollars, is given by the function
a. What price range would yield a profit for this item?
b. Describe what happens to the profit as the price increases. Why is a quadratic function an appropriate model for profit as a function of price?
c. What price would yield a maximum profit?
29. A dairy farmer has 1500 feet of fencing. He wants to use all 1500 feet to construct a rectangle and two interior separators that together form three rectangular pens. See the accompanying figure.
a. If W is the width of the larger rectangle, express the length, L, of the larger rectangle in terms of W.
b. Express the total area, A(W), of the three pens as a polynomial in terms of W.
c. What is the domain of the function A(W)?
d. What are the dimensions of the larger rectangle that give a maximum area? What is the maximum area?
7. A worker gets $20/hour for a normal work week of 40 hours and time-and-a-half for overtime. Assuming he works at least 40 hours a week, construct a function describing his weekly paycheck as a function of the number of hours worked.
13. The Richland Banquet Hall charges $500 to rent its facility and $40 per person for dinner. The Hall rental requires a minimum of 25 people and a maximum of 100. A sorority decides to hold its formal there, splitting all the costs among the attendees. Let n be the number of people attending the formal.
a. Create a function C(n) for the total cost of renting the hall and serving dinner.
b. Create a function P(n) for the cost per person for the event.
c. What is P(25)? P(100)? What do these numbers represent?
10. Estimate the maximum number of turning points for each of the polynomial functions. If available, use technology to graph the function to verify the actual number.
a. y = x4 – 2x2 – 5 c. y = x3 – 3x2 + 4
b. y = 4t6 + t2 d. y = 5 + x
28. An ice bucket is designed as a cubic block of foam with a centered cylindrical ice cavity. The side of the outer cube is s inches. Because there is 1² minimum thickness of foam at each side of the cube, the diameter of the cylindrical cavity is s – 2. The bottom of the cavity is 1² from the bottom of the cube. There is no lid.
20. A student drives non-stop from Missoula, Montana to Spokane, Washington. The trip consists of 110 miles of travel in Montana and 90 miles of travel in Idaho and Washington. The speed limit is 75 mph in Montana and is 65 in Idaho and Washington.
a. If the student drives at the speed limit (with no stops), how long would the trip take?
b. Construct an expression for T(x) which represents the driving time if the students drives x mph over the speed limit. Rewrite T(x) as the quotient of two polynomials.
c. Find T(0) and compare with your answer in part (a).
d. Find T(10). Is the driving time shorter or longer than T(0)? Why?
e. Find T(–5). Is the driving time shorter or longer than T(0)? Why?
7. The winds are calm, allowing a forest fire to spread in a circular fashion at 5 feet per minute.
a. Construct a function A(r) for the circular area burned, where r is the radius. Identify the units for the input and the output of A(r).
b. Construct a function for the radius r = R(t) for the increase in the fire radius as a function of time t. What are the units now for the input and the output for R(t)?
c. Construct a composite function that gives the burnt area as a function of time. What are the units now for the input and the output?
d. How much forest area is burned after 10 minutes? One hour?
8. The exchange rate a bank gave for Canadian dollars on June 27, 2010, was 0.961 Canadian dollars for 1 U.S. dollar. The bank also charges a constant fee of 3 U.S. dollars per transaction.
a. Construct a function F that converts U.S. dollars, d, to Canadian dollars.
b. Construct a function G that converts Canadian dollars, c, to U.S. dollars.
c. What would the function F ° G do? Would its input be U.S. or Canadian dollars (i.e., d or c)? Construct a formula for F ° G.
d. What would the function G ° F do? Would its input be U.S. or Canadian dollars (i.e., d or c)? Construct a formula for G ° F.
9. A stone is dropped into a pond, causing a circular ripple that is expanding at a rate of 13 ft/s. Describe the area of the circle as a function of time.
10. The wind chill temperature is the apparent temperature caused by the extra cooling from the wind. A rule of thumb for estimating the wind chill temperature for an actual temperature t that is above 0o Fahrenheit is W(t) = t – 1.5S0, where S0 is any given wind speed in miles per hour.
a. If the wind speed is 25 mph and the actual temperature is 10° F, what is the wind chill temperature?
b. Construct a function that will give the wind chill temperature as a function of degrees Celsius.
c. If the wind speed is 40 mph and the actual temperature is –10° C, what is the wind chill temperature?
11. Salt is applied to roads to decrease the temperature at which icing occurs. Assume that with no salt, icing occurs at 32° F, and that each unit increase in the density of salt applied decreases the icing temperature by 5° F.
a. Construct a formula for icing temperature, T(s), as a function of salt density, s.
Trucks spread salt on the road, but they do not necessarily spread it uniformly across the road surface. If the edges of the road get half as much salt as the middle, we can describe salt density s = S(x) as a function of the distance, x, from the center of the road by where k is the distance from the centerline to the road edges and , where k is the salt density applied in the middle of the road.
b. What will the expression for S(x) be if the road is 40 feet wide?
c. What will the value for x be at the middle of the 40-footwide road? At the edge of the road? Verify that at the middle of the road the value of the salt density S(x) is S d and that at the edge the value of S(x) is .
d. Construct a function that describes the icing temperature, T, as a function of x, the distance from the center of the 40-foot-wide road.
e. What is the icing temperature at the middle of the 40-foot-wide road? At the edge?
25. Cryptology (the creation and deciphering of codes) is based on 1-1 functions. After you code a message using a 1-1 function, the decoder needs the inverse function in order to retrieve the original message. The following table matches each letter of the alphabet with its coded numerical form.
a. Does this code represent a 1-1 function? Is there an inverse function? If so, what is its domain?
b. Decode the message “14 26 7 19 9 6 15 22 8.”
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