c. Solve the system of two equations to find the speed of the boat in the water (x) and the speed of the current (y). (0.45, 0.05)

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1 Math Applications The applications that follow are like the ones you will encounter in many workplaces. Use the mathematics you have learned in this chapter to solve the problems. Wherever possible, use your calculator to solve the problems that require numerical answers. 1 A motorboat takes 40 minutes to travel 20 miles down a river with the help of the current. On the return trip, the boat is powered at the same level and moves against the current. The upstream trip (20-miles) takes 50 minutes. a. Let x represent the speed of the boat in the water and y represent the speed of the water current. While traveling downstream, the boat s speed as seen from the shore is the sum of the speed of the current and the speed of the boat in the water. This sum is equal to the speed of 20 miles per 40 minutes, or 20/40 miles per minute. Write an equation that describes the downstream speed. x y 0.5 b. While traveling upstream, the boat s speed as seen from the shore is the difference between the speed of the current and the speed of the boat in the water. This difference is equal to the speed of 20 miles per 50 minutes, or 20/50 miles per minute. Write an equation that describes the upstream speed. x y 0.4 c. Solve the system of two equations to find the speed of the boat in the water (x) and the speed of the current (y). (0.45, 0.05) d. What are the units for the speeds you determined: Miles per hour? Miles per minute? Feet per minute? Explain. miles per minute; see margin for explanation 2 You and your friends are planning a kayak trip. You are in charge of deciding which kayak rental company will be the least expensive. Company A charges $5 per hour for the use of its kayaks. Company B charges an $8 usage fee plus $3 per hour. If c represents the total cost of using the kayaks for h hours, you can write the following equations for each company. Company A: c 5h Company B: c 3h 8 a. Draw a graph for each company showing cost as a function of hours. see margin b. Find the number of hours where the costs of the two plans are equal. 4 c. For a 6-hour trip, which plan is less expensive? Company B 480 Chapter 8 Systems of Equations

2 3 Bill and Laura mow lawns during the summer to earn money. They each calculate their start-up expenses, operating expenses, and income per hour of mowing. Bill and Laura write the following equations for their total income (y) after mowing for x hours. Bill: y 9.10x 125 Laura: y 9.10x 100 a. Graph each equation on the same coordinate plane. Use x-values from zero to 20 hours. see margin b. Before Bill and Laura each mow their first lawn, what is their total income? Explain your answer. Bill is $( 125) and Laura is $( 100). c. After mowing for 20 hours, what is each of their total incomes? Bill s is $57 and Laura s is $82. d. If both Bill and Laura work the same number of hours, when will Bill have the same total income as Laura? Explain your answer. Laura will always make more than Bill; The lines are parallel. 4 A farmer plants two kinds of crops on his 2,500 acres of land. The income from Crop A is $230 per acre. The income from Crop B is $280 per acre. The farmer s goal is to earn $625,000 from the sale of the crops. a. Let x represent the number of acres of Crop A and y represent the number of acres of Crop B. Equate sales of x acres of Crop A plus y acres of Crop B to the desired total sales of $625, x 280y 625,000 b. If the farmer uses all 2,500 acres of available land, write an equation for the total number of acres planted. x y 2,500 c. Solve this system of equations to find the number of acres the farmer should plant of each type of crop to obtain the desired total sales. (1,500, 1,000); 1,500 acres at $230; 1,000 acres at $280 d. Check your result by substituting the x- and y-values into your equations. 230(1,500) 280(1,000) 625,000; 1, , ,500 Math Applications 481

3 5 You have been offered a job marketing grain products. The company has two salary plans. Plan A pays a weekly salary of $700 plus a commission of $50 for each ton of grain products that you sell. Plan B pays a weekly salary of $500 plus a commission of $70 for each ton sold. a. Write these two payment plans as a system of equations. y 50x 700; y 70x 500 b. Draw a graph for each equation. see margin c. How many tons of grain must you sell for Plan A and Plan B to pay the same amount? 10 d. You estimate that you can sell about 25 tons of grain products per week. Which plan should you choose? Plan B 6 Mario is stocking a new pond for his fish farm. The pond is filled by a springfed creek at a rate that can support F fish F 42m 250, where m is the number of months the pond has been filling. Mario stocks the pond with an initial population of 110 fish (at month m 0). He predicts a population growth rate of 8% per month. Using this information, Mario writes an equation for the population size P P 110(1.08) m, where P is the number of fish in the pond and m is the number of months since the fish were placed in the pond. Notice that this equation is not linear. This problem can be solved by graphing. a. Graph both equations on a graphing calculator. Show values for m from 0 months up to 50 months. see margin b. Mario predicts that the pond will take about 40 months to fill. He wants to begin harvesting the fish just before the population exceeds the limit that the pond can support. When will the predicted population of fish (P) equal the number of fish the pond can support (F )? about 36 months c. Should Mario wait till the pond fills to begin harvesting the fish, or should he start sooner? start sooner 482 Chapter 8 Systems of Equations

4 7 Chris is in charge of buying new suits for a men s shop. The purchasing budget is $20,000. Chris decides to stock the inventory with two different types of suits. One type is a designer label costing $400. The other is a nondesigner label costing $250. From previous sales records, Chris knows customers will buy nondesigner suits about 60% of the time and designer suits about 40% of the time. a. Let x represent the number of nondesigner suits and y represent the number of designer suits. Equate the purchasing budget of $20,000 to the cost of purchasing x nondesigner suits and y designer suits. 250x 400y 20,000 b. To meet the sales demand, 60% of the purchases should be nondesigner suits and 40% should be designer suits. Write this as a proportion and solve the proportion for y. x y ; y x c. Solve the system of equations from parts a and b by substitution. (38.71, 25.81) d. How many of each type suit should Chris purchase to stay within his budget and meet the sales demand? 38 nondesigner and 26 designer or 40 nondesigner and 25 designer e. Check your result by substituting the x- and y-values into your equations. 250(38.71) 400(25.81) 20,000; (38.71) 8 You are starting a business that sells water filters. Your initial start-up cost is $1,500, and each filter kit will cost $50. You plan to sell each unit for $75. From this information, you can write the following equations, where n is the number of filters sold. Cost 50n 1,500 Income 75n a. Write each equation as a function of n. Draw the graph of each function on the same coordinate plane. see margin b. From the graph, determine how many filters you need to sell to break even. (That is, how many filters do you need to sell for income to equal cost?) 60 Math Applications 483

5 9 Susan s new company jet carries enough fuel for eight hours of flying when cruising at an airspeed of 220 miles per hour. Susan is heading west with a tailwind of 15 miles per hour. She needs to find the maximum distance west she can fly and still have enough fuel left to return home. a. Consider the following variables: the time t w for the westerly (outbound) flight and the time t e for the easterly (return) flight. The total flight time is limited by the amount of fuel. Write an equation that shows the flight times, t w and t e, totaling eight hours. t w t e 8 b. Because the westerly flight is assisted by a tailwind, the airplane s speed is miles per hour. The distance traveled during the westerly flight is this speed times t w. Write the product for the distance traveled during the westerly flight. D w 235t w c. Because the easterly flight is into a headwind, the airplane s speed is miles per hour. The distance traveled during the easterly flight is this speed times t e. Write the product for the distance traveled during the easterly flight. D e 205t e d. Of course, Susan wants the outbound westerly flight distance to equal the easterly return flight distance. Equate the two products from parts b and c to obtain a second equation containing the variables t w and t e. 235t w 205t e e. Solve the system of two equations. Check your results by substituting into your original equations. t w 3.7; t e 4.3 f. Use the equation from part b to find how far west Susan can fly under these conditions and still have enough fuel to return home. about miles 10 Your trucking company needs to move 21 tons of gravel. You have eight qualified drivers in the company and two types of trucks. One type of truck can haul 5 tons and the other type can haul 3 tons. An insurance requirement specifies that 5-ton trucks must have two drivers in the cab during operation. Three-ton trucks require only one driver. Let x be the number of 5-ton trucks you will use and y the number of 3-ton trucks. a. Write two linear equations relating x and y that determine how many of each size truck are needed to move the gravel in one trip using all available drivers. 2x y 8; 5x 3y 21 b. Solve the system of equations by multiplication. (3, 2) c. How many of each size truck will you use? 3 of the 5-ton; 2 of the 3-ton Westerly speed = 220 mph + 15 mph Easterly speed = 220 mph 15 mph 484 Chapter 8 Systems of Equations

6 11 You must determine the price schedule for concert seating at the local orchestra hall. You will sell tickets for two types of seats: one type will sell for $5 each, and the second type for $8 each. There are a total of $5 1,500 seats, and you expect to sell tickets for all of them. Your ticket sales total $10,500. Let x $5 $5 be the number of seats that sell for $8 each and y be the number of seats that sell for $5 each. $8 $8 a. Write an equation for the total number of seats as the sum of the two different types of seats. x y 1,500 Orchestra Stage b. Write an equation for the income from the sale of tickets ($10,500). 8x 5y 10,500 c. Solve this system of equations. (1,000, 500) 12 Amad wants to earn $500 on a $6,000 investment. He is going to split his investment between stocks that he predicts will have an annual yield of 10% and a mutual fund he predicts will have a 7% annual yield. Let x represent the amount invested in stocks and y be the amount invested in the mutual fund. a. Write the equation for the total amount invested as the sum of the amount invested in stocks (x) and the amount invested in the mutual fund (y). x y 6,000 b. Write an equation that equates the amount that Amad wants to earn with the sum of the yield from the stocks (10% of the amount invested in stock) and the yield from the mutual fund (7% of the amount invested in the mutual fund). 0.10x 0.07y 500 c. Graph both equations on the same coordinate axes. see margin d. From the graph, approximate the amount that Amad should invest in stocks and the amount he should invest in the mutual fund. about $2,667 in stock and $3,333 in mutual fund Math Applications 485

7 13 You are in charge of catering a banquet. To keep the costs down, you will serve only two entrees. One is a chicken dish that costs $5; the other is a beef dish that costs $7. The banquet will have 250 people, and the total cost of the food is $1,500. a. Let x be the number of chicken dishes you will prepare. Let y be the number of beef dishes. Equate the total number of entrees to the total number of people. x y 250 b. Write an equation that equates the total cost of the food to the cost for all the chicken dishes plus the cost for all the beef dishes. 5x 7y 1,500 c. Find the number of each type of entree you should prepare. Check your answers. 125 beef; 125 chicken 14 You are redecorating your home. You have decided to put down a combination of carpet and vinyl floor covering in the family room. The carpet costs $2 per square foot, and the vinyl covering costs $1 per square foot. You can spend a total of $500 on the materials. The area you want to cover is 300 square feet. a. Let x be the number of square feet of carpet you will use. Let y be the number of square feet of vinyl. Equate the total cost of the materials to the cost of the carpet and the vinyl. 2x y 500 b. Write an equation that equates the total floor area (300 square feet) to the sum of the area covered by carpet and the area covered by vinyl. x y 300 c. Solve this system of equations to find the number of square feet of carpet and vinyl that meet your requirements. Check your answers. 200 square feet of carpet; 100 square feet of vinyl 15 You are shopping for refrigerators. Brand A costs $600 and uses $68 per year in electricity. Brand B costs $900 but uses only $55 per year in electricity. a. Write a system of equations to model the total cost of each refrigerator over t years. A 68t 600; B 55t 900 b. Assume that the domain of each equation is from zero years to 15 years (why not 70 years?). Draw a graph of the system. see margin c. Over the lifetime (that is, the domain) of these refrigerators, will the total costs ever be the same? If so, when? From zero to 15 years, there is no solution. d. Which refrigerator has a lower total cost over a 15-year lifetime? A 486 Chapter 8 Systems of Equations

8 16 As an interior decorator, you are selecting the lighting for a client s new family room. You must decide between regular incandescent light bulbs and energy-saving fluorescent light bulbs. The incandescent bulbs cost 50 each and will last for 1,000 hours. The fluorescent light bulbs cost $11 each and last 9,000 hours. However, the incandescent bulb costs $0.006 per hour to operate, while the fluorescent bulb operates for only $ per hour. You want to show your client that you have chosen the more economical lighting system. a. Write a system of equations for the total cost of each type of light bulb for equal time periods. (Hint: one fluorescent bulb lasts as long as nine incandescent bulbs.) CI 0.006x (0.5)9; CF x 11 b. Graph the two equations for the total cost for each bulb over the span of 9,000 hours. see margin c. Is there a solution to these two equations? That is, will the total cost of the incandescent bulb ever equal the total cost of the fluorescent bulb? If so, when? yes; $13.79 at 1,548 hours d. Based on cost alone, which bulb would you recommend to your client? Explain your reasoning. fluorescent bulbs 17 Shon needs three liters of an 8% saline solution. He has a 5% saline solution and a 9% solution in the lab stock room. Before he mixes the two solutions, he needs to calculate the right proportions. Let x be the number of liters of the 5% solution and y be the number of liters of the 9% solution. a. Equate the sum of x and y to the total number of liters needed. x y 3 b. To obtain a second equation, Shon starts with the word equation below. The equation uses the fact that the amount of salt in the saline solution is the concentration of the solution times the number of liters of solution. When Shon mixes solution A and solution B, the amount of salt in the mixture is the amount of salt from solution A plus the amount of salt from solution B. (Concentration ofa)(liters of A) (Concentration of B)(Liters of B) (Concentration of mixture)(liters of mixture) Substitute the appropriate variables and numbers into the word equation and write Shon s second linear equation. 0.05x 0.09y 0.8(3) c. Use the method of substitution to solve the system of two equations. How many liters of 5% and 9% saline solution should Shon mix to obtain 3 liters of 8% saline solution? Check your answers liters of 5%; 2.25 liters of 9% Math Applications 487

9 18 Fahrenheit and Celsius are two different scales of temperature measurement. Linear equations relate the two scales F ac b, where F is the temperature in degrees Fahrenheit, C is the temperature in degrees Celsius, and a and b are the constants to be determined. The two scales are related by the fact that water freezes at 32 F or 0 C. Similarly, water boils at 212 F or 100 C. a. Substitute the temperatures for freezing water into the equation above to obtain an equation with unknowns a and b. 32 a(0) b b. Substitute the temperatures for boiling water into the equation above to obtain a second equation with unknowns a and b. 212 a(100) b c. Solve this system of equations to determine the values of a and b that satisfy both equations. (1.8, 32) d. Rewrite the equation F ac b with the calculated values of a and b. Check the resulting equation for C 0 and C 100. F 1.8C Laura needs 4,000 pounds of a 17% copper alloy. To make the alloy, she will mix a 23% copper alloy and a 12% copper alloy. Let x be the number of pounds of the 23% copper alloy. Let y be the number of pounds of the 12% copper alloy. a. Write the equation for the total number of pounds needed for the mixture as the sum of the pounds of 12% alloy and the pounds of 23% alloy. x y 4,000 b. To obtain a second equation, Laura starts with the word equation below. The equation uses the fact that the amount of copper in an alloy is the concentration of the alloy times the number of pounds of alloy. When Laura mixes the two alloys, the amount of copper in the mixture is the amount of copper from one alloy plus the amount of copper from the other alloy. (Concentration of A)(Quantity of A) (Concentration of B)(Quantity of B) (Concentration of mixture)(quantity of mixture) Rewrite this word equation using the variables x and y. Remember to express the percentages as decimal values. 0.23x 0.12y Chapter 8 Systems of Equations

10 c. Use the method of substitution to solve the two equations. How many pounds of 23% and 12% alloy should Laura mix to obtain 4,000 pounds of 17% alloy? Check your answers by substituting the results for x and y into your equations. 1,818 pounds of 23% alloy; 2,182 pounds of 12% alloy 20 You are calibrating a new thermocouple that is used for measuring temperatures between 0 C and 100 C. To do this, you must be able to relate the thermocouple voltage to the temperature being measured. You know that the thermocouple s output voltage is roughly linear between 0 C and 100 C (that is, you can write a calibration equation in the form y = mx + b). Thus, for a temperature T, the thermocouple should produce a voltage V according to this calibration equation. T Vm b Two of the test measurements with your new thermocouple show that when T 0 C, V 1.56 millivolts, and when T 100 C, V 4.76 millivolts. a. Substitute each pair of temperature and voltage measurements into the calibration equation to obtain equations with unknowns m and b. 0 ( 1.56)m b; m 1 b b. Solve for m and b. Check the results by substituting your answers into the two equations. (15.82, 24.68) c. Write the correct calibration equation with the values of m and b you determined. T 15.82V Kirchhoff s loop rule analyzes the current (I) in an electronic circuit. This rule states that in any closed loop of a circuit, the sum of the voltage drops (across the resistors) must equal the sum of the voltages supplied. If you apply this rule to the circuit shown here, you obtain the following equations I 1 2 7(I 1 I 2 ) I 2 7(I 1 I 2 ) 0 12 Ω 7 Ω 12 V 16 V I 1 I 2 a. Simplify each equation to obtain a system of equations in standard form. 19I 1 7I 2 12; 7I 1 29I Ω b. Solve this system of equations for I 1 and I 2. Check your solution. The currents should be about amperes and amperes. Math Applications 489