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1.2 Linear Applications – Perimeter, Mixture, & Investment Problems 1-1 Linear Applications Formulas (These need to be memorized!) Perimeter Prectangle = 2l + 2w = 2(l + w) Psquare = 4s Ccircle = 2r = d Area Arectangle = lw Asquare = s2 Acircle = r 2 Distance d = r t Simple Interest I = prt, p = principle r = rate t = time 1-2 Find the Dimensions of a Square The length of a rectangle is 2 in. more than the width. If the length and width are each increased by 3 in., the perimeter of the new rectangle will be 4 in. less than 8 times the width of the original rectangle. Find the dimensions of the original rectangle. Assign variables: Let x = the length of the original rectangle. Then, x − 2 = the width of the original rectangle. x + 3 = the length of the new rectangle (x − 2) + 3 = x + 1 = the width of the new rectangle. 1-3 Find the Dimensions of a Square (cont.) The perimeter of the new rectangle is The perimeter of the new rectangle is 4 in. less than 8 times the width of the original rectangle, so we have 1-4 Find the Dimensions of a Square (cont.) Distributive property. Combine terms. Add –4x and 20 to both sides. Divide both sides by 4. The length of the original rectangle is 7 in. The width of the original rectangle is 7 – 2 = 5 in. Be sure to include units (when given) in your answer! 1-5 Find the Dimensions of a Square If the length of each side of a square is increased by 3 cm, the perimeter of the new square is 40 cm more than twice the length of each side of the original square. Find the dimensions of the original square. Assign variables: Let x = the length of each side of the original square. Then, x + 3 = the length of each side of the new square. The perimeter of the new square is p = 4(x + 3). 1-6 1.2 Example 1 Find the Dimensions of a Square (cont.) The perimeter of the new square is also given as p = 2x + 40 These two quantities must be equal: 4(x + 3) = 2x + 40 4x + 12 = 2x + 40 2x = 28 x = 14 Each side of the original square measures 14 cm. 1-7 College Algebra K/DC Monday, 11 November 2013 • OBJECTIVE TSW (1) work with parallel and perpendicular lines, and (2) solve linear application problems. • ASSIGNMENTS DUE people who were absent Friday – Sec. 1.1: p. 89 (39-47 all, 49-58 all) give to me – WS Slopes and Intercepts give to me • ASSIGNMENTS DUE – WS Equations of Lines wire basket – WS Parallel and Perpendicular black tray 8 Solving a Mixture Problem How many gallons of a 25% anti-freeze solution should be added to 5 gallons of a 10% solution to obtain a 15% solution? Let x = the amount of 25% solution The number of gallons of pure antifreeze in the 25% solution plus the number of gallons of pure antifreeze in the 10% solution must equal the number of gallons of pure antifreeze in the 15% solution. 1-9 Solving a Mixture Problem (cont.) Create a table to show the relationships in the problem. Write an equation: 1-10 Solving a Mixture Problem (cont.) Distributive property. Subtract .15x and .5. Divide by .1. 2.5 gallons of the 25% solution should be added. 1-11 Solving a Mixture Problem Charlotte Besch is a chemist. She needs a 20% solution of alcohol. She has a 15% solution on hand, as well as a 30% solution. How many liters of the 15% solution should she add to 3 L of the 30% solution to obtain her 20% solution? Let x = the amount of 15% solution The number of liters in the 15% solution plus the number of liters in the 30% solution must equal the number of liters in the 20% solution. 1-12 Solving a Mixture Problem (cont.) Create a table to show the relationships in the problem. Strength L of Solution L of Pure Alcohol 15% x 0.15x 30% 3 0.30(3) 20% 3+x 0.20(3 + x) Write an equation: 0.15x + 0.30(3) = 0.20(3 + x) 1-13 1.2 Example 2 Solving a Mixture Problem (cont.) 0.15x + 0.30(3) = 0.20(3 + x) 0.15x + 0.90 = 0.60 + 0.20x 0.30 = 0.05x 6=x 6 liters of the 15% solution should be added. 1-14 Solving an Investment Problem Last year, Owen earned a total of $1456 in interest from two investments. He invested a total of $28,000, part at 4.8% and the rest at 5.5%. How much did he invest at each rate? Let x = amount invested at 4.8%. Then 28,000 − x = amount invested at 5.5%. 1-15 Solving an Investment Problem Create a table to show the relationships in the problem. The amount of interest from the 4.8% account plus the amount of interest from the 5.5% account must equal the total amount of interest. 1-16 Solving an Investment Problem Distributive property Combine terms. Subtract 1540. Divide by –.007. Owen invested $12,000 at 4.8% and $28,000 − $12,000 = $16,000 at 5.5%. 1-17 Solving a Motion Problem Example 1 Krissa drove to her grandmother’s house. She averaged 40 mph driving there. She was able to average 48 mph returning, and her driving time was 1 hr less. What is the distance between Krissa’s house and her grandmother’s house? Set up a chart showing the relationships. r t d To Grandma 40 x 40x To Home 48 x–1 48(x – 1) 1-18 Solving a Motion Problem Example 1 r t d To Grandma 40 x 40x To Home 48 x–1 48(x – 1) The distance is the same both going to and coming back: 40x = 48(x – 1) 40x = 48x – 48 48 = 8x 6=x 1-19 Solving a Motion Problem Example 1 Now use this to answer the question: What is the distance between Krissa’s house and her grandmother’s house? 40(6) = 240 miles 1-20 Solving a Motion Problem Example 2 Maria and Eduardo are traveling to a business conference. The trip takes 2 hr for Maria and 2.5 hr for Eduardo, since he lives 40 mi farther away. Eduardo travels 5 mph faster than Maria. Find their average rates. Set up a chart: Maria Eduardo r t d x 2 2x x+5 2.5 2.5(x + 5) 1-21 Solving a Motion Problem Example 2 Maria Eduardo r t d x 2 2x x+5 2.5 2.5(x + 5) Are the two distances the same? NO Eduardo lives 40 mi further, so 2x + 40 = 2.5(x + 5) 2x + 40 = 2.5x + 12.5 27.5 = 0.5x x = 55 Maria’s rate is 55 mph; Eduardo’s rate is 60 mph. 1-22 Assignment • Sec. 1.2: pp. 97-98 (5-10 all, 13-17 all) • Due on Friday, 15 November 2013 (TEST day). • Sec. 1.2: pp. 99-100 (19-33 odd, 37, 38) • You do not need to write the problem, but you do need to show work. • Due on Friday, 15 November 2013 (TEST day). 1-23 Assignment: Sec. 1.2: pp. 99-100 (19-33 odd, 37, 38) 19) In the morning, Margaret drive to a business appointment at 50 mph. Her average speed on the return trip in the afternoon was 40 mph. The return trip took ¼ hr longer because of heavy traffic. How far did she travel to the appointment? 21) David gets to work in 20 min when he drives his car. Riding his bike (by the same route) takes him 45 min. His average driving speed is 4.5 mph greater than his average speed on his bike. How far does he travel to work? 23) Russ and Janet are running in the Strawberry Hill Fun Run. Russ runs at 7 mph, Janet at 5 mph. If they start at the same time, how long will it be before they are 1.5 mi apart? 1-24 Assignment: Sec. 1.2: pp. 99-100 (19-33 odd, 37, 38) 25) On September 14, 2002, Tim Montgomery (USA) set a world record in the 100-m dash with a time of 9.78 sec. If this pace could be maintained for an entire 26-mi marathon, what would his time be? How would this time compare to the fastest time for a marathon of 2 hr, 5 min, 38 sec? (Hint: 1 m ≈ 3.281 ft) 27) Joann took 20 min to drive her boat upstream to water-ski her favorite spot. Coming back later in the day, at the same boat speed, took her 15 min. If the current in that part of the river is 5 km per hr, what was her boat speed? 29) How many gallons of 5% acid solution must be mixed with 5 gal of 10% solution to obtain a 7% solution? 1-25 Assignment: Sec. 1.2: pp. 99-100 (19-33 odd, 37, 38) 31) Beau Glaser wishes to strengthen a mixture from 10% alcohol to 30% alcohol. How much pure alcohol (100%) should be added to 7 L of the 10% mixture? 33) How much water (0%) should be added to 8 mL of 6% saline solution to reduce the concentration to 4%? 37) In planning her retirement, Callie Daniels deposits some money at 2.5% interest, with twice as much deposited at 3%. Find the amount deposited at each rate if the total annual interest income is $850. 39) Linda won $200,000 in a state lottery. She first paid income tax of 30% on the winnings. Of the rest she invested some at 1.5% and some at 4%, earning $4350 interest per year. 1-26 How much did she invest at each rate?