2.3 Formulas and Problem Solving

Transcription

1 Section.3 Formulas and Problem Solving 69.3 Formulas and Problem Solving OBJECTIVES 1 Solve a Formula for a Specified Variable. Use Formulas to Solve Problems. OBJECTIVE 1 Solving a Formula for a Specified Variable Solving problems tat we encounter in te real world sometimes requires us to express relationsips among measured quantities. An equation tat describes a known relationsip among quantities suc as distance, time, volume, weigt, money, and gravity is called a formula. Some examples of formulas are Formula Meaning I = PRT Interest = principal # rate # time A = lw Area of a rectangle = lengt # widt d = rt Distance = rate # time C = pr Circumference of a circle = # p # radius V = lw Volume of a rectangular solid = lengt # widt # eigt Oter formulas are listed in te front cover of tis text. Notice tat te formula for te volume of a rectangular solid V = lw is solved for V since V is by itself on one side of te equation wit no V s on te oter side of te equation. Suppose tat te volume of a rectangular solid is known as well as its widt and its lengt, and we wis to find its eigt. One way to find its eigt is to begin by solving te formula V = lw for. EXAMPLE 1 Solve: V = lw for. Solution To solve V = lw for, isolate on one side of te equation. To do so, divide bot sides of te equation by lw. V = lw V lw = lw lw Divide bot sides by lw. V lw = or = V lw Simplify. Tus we see tat to find te eigt of a rectangular solid, we divide its volume by te product of its lengt and its widt. 1 Solve: I = PRT for T. Te following steps may be used to solve formulas and equations in general for a specified variable. Solving Equations for a Specified Variable Step 1. Clear te equation of fractions by multiplying eac side of te equation by te least common denominator. Step. Use te distributive property to remove grouping symbols suc as parenteses. Step 3. Combine like terms on eac side of te equation. Step 4. Use te addition property of equality to rewrite te equation as an equivalent equation wit terms containing te specified variable on one side and all oter terms on te oter side. Step 5. Use te distributive property and te multiplication property of equality to isolate te specified variable.

2 70 CHAPTER Equations, Inequalities, and Problem Solving EXAMPLE Solve: 3y - x = 7 for y. Solution Tis is a linear equation in two variables. Often an equation suc as tis is solved for y to reveal some properties about te grap of tis equation, wic we will learn more about in Capter 3. Since tere are no fractions or grouping symbols, we begin wit Step 4 and isolate te term containing te specified variable y by adding x to bot sides of te equation. 3y - x = 7 3y - x + x = 7 + x Add x to bot sides. 3y = 7 + x To solve for y, divide bot sides by 3. 3y 3 = 7 + x 3 y = x Solve: 7x - y = 5 for y. Divide bot sides by 3. or y = x EXAMPLE 3 Solve: A = 1 1B + b for b. Solution Since tis formula for finding te area of a trapezoid contains fractions, we begin by multiplying bot sides of te equation by te LCD. Helpful Hint Remember tat we may get te specified variable alone on eiter side of te equation. b B A = 1 1B + b # A = # 1 1B + b Multiply bot sides by. A = 1B + b Simplify. Next, use te distributive property and remove parenteses. A = 1B + b A = B + b Apply te distributive property. A - B = b Isolate te term containing b by subtracting B from bot sides. Divide bot sides by. A - B A - B = b = b or b = 3 Solve: A = P + Prt for r. A - B OBJECTIVE Using Formulas to Solve Problems In tis section, we also solve problems tat can be modeled by known formulas. We use te same problem-solving steps tat were introduced in te previous section.

3 Section.3 Formulas and Problem Solving 71 Formulas are very useful in problem solving. For example, te compound interest formula A = Pa1 + r nt n b is used by banks to compute te amount A in an account tat pays compound interest. Te variable P represents te principal or amount invested in te account, r is te annual rate of interest, t is te time in years, and n is te number of times compounded per year. EXAMPLE 4 Finding te Amount in a Savings Account Karen Estes just received an ineritance of $10,000 and plans to place all te money in a savings account tat pays 5% compounded quarterly to elp er son go to college in 3 years. How muc money will be in te account in 3 years? Solution 1. UNDERSTAND. Read and reread te problem. Te appropriate formula to solve tis problem is te compound interest formula A = Pa1 + r nt n b Make sure tat you understand te meaning of all te variables in tis formula. A = amount in te account after t years P = principal or amount invested t = time in years r = annual rate of interest n = number of times compounded per year. TRANSLATE. Use te compound interest formula and let P = +10,000, r = 5, = 0.05, t = 3 years, and n = 4 since te account is compounded quarterly, or 4 times a year. Formula: A = Pa1 + r nt n b Substitute: A = 10,000a b 4# 3 3. SOLVE. We simplify te rigt side of te equation. A = 10,000a b 4# 3 A = 10, A 10, A 11, Simplify and write 4 # 3 as 1. Approximate Multiply and round to two decimal places. 4. INTERPRET. Ceck: Repeat your calculations to make sure tat no error was made. Notice tat $11, is a reasonable amount to ave in te account after 3 years. State: In 3 years, te account will contain $11, Russ placed $8000 into is credit union account paying 6% compounded semiannually (twice a year). How muc will be in Russ s account in 4 years?

4 7 CHAPTER Equations, Inequalities, and Problem Solving Graping Calculator Explorations To solve Example 4, we approximated te expression 10,000a # 3 4 b Use te keystrokes sown in te accompanying calculator screen to evaluate tis expression using a graping calculator. Notice te use of parenteses. EXAMPLE 5 Finding Cycling Time Te fastest average speed by a cyclist across te continental United States is 15.4 mp, by Pete Penseyres. If e traveled a total distance of about miles at tis speed, find is time cycling. Write te time in days, ours, and minutes. (Source: Te Guinness Book of World Records) Solution 1. UNDERSTAND. Read and reread te problem. Te appropriate formula is te distance formula d = rt were d = distance traveled r = rate and t = time. TRANSLATE. Use te distance formula and let d = miles and r = 15.4 mp. 3. SOLVE = 15.4t t d = rt = 15.4t Divide bot sides by Te time is approximately ours. Since tere are 4 ours in a day, we divide by 4 and find tat te time is approximately 8.41 days. Now, let s convert te decimal part of 8.41 days back to ours. To do tis, multiply 0.41 by 4 and te result is 9.84 ours. Next, we convert te decimal part of 9.84 ours to minutes by multiplying by 60 since tere are 60 minutes in an our. We ave 0.84 # minutes rounded to te nearest wole. Te time is ten approximately 8 days, 9 ours, 50 minutes. 4. INTERPRET. Ceck: Repeat your calculations to make sure tat an error was not made. State: Pete Penseyres s cycling time was approximately 8 days, 9 ours, 50 minutes. 5 Nearly 4800 cyclists from 36 U.S. states and 6 countries rode in te Pan- Massacusetts Callenge recently to raise money for cancer researc and treatment. If te riders of a certain team traveled teir 19-mile route at an average speed of 7.5 miles per our, find te time tey spent cycling. Write te answer in ours and minutes.

5 Section.3 Formulas and Problem Solving 73 Vocabulary, Readiness & Video Ceck Solve eac equation for te specified variable. See Examples 1 troug x + y = 5 for y. 7x - y = 3 for y 3. a - 5b = 8 for a 4. 7r + s = 10 for s 5. 5j + k - = 6 for k 6. w - 4y + z = 0 for z Martin-Gay Interactive Videos Watc te section lecture video and answer te following questions. OBJECTIVE 1 OBJECTIVE 7. Based on te lecture before Example 1, wat two tings does solving an equation for a specific variable mean? 8. As te solution is cecked at te end of Example 3, wy do you tink it is mentioned to be especially careful tat you use te correct formula wen solving problems? See Video.3.3 Exercise Set Solve eac equation for te specified variable. See Examples d = rt; for t. W = g; for g 3. I = PRT; for R 4. V = lw; for l 5. 9x - 4y = 16; for y 6. x + 3y = 17; for y 7. P = L + W; for W 8. A = 3M - N; for N 9. J = AC - 3; for A 10. y = mx + b; for x 11. W = g - 3gt ; for g 1. A = Prt + P; for P 13. T = C1 + AB; for B 14. A = 5H1b + B; for b 15. C = pr; for r 16. S = pr + pr; for 17. E = I1r + R; for r 18. A = P11 + rt; for t 19. s = n 1a + L; for L 0. C = 5 1F - 3; for F 9 1. N = 3st 4-5sv; for v. L = a + 1n - 1d; for d 3. S = LW + LH + WH; for H 4. T = 3vs - 4ws + 5vw; for v In tis exercise set, round all dollar amounts to two decimal places. Solve. See Example Complete te table and find te balance A if $3500 is invested at an annual percentage rate of 3% for 10 years and compounded n times a year. n A 6. Complete te table and find te balance A if $5000 is invested at an annual percentage rate of 6% for 15 years and compounded n times a year. n A 7. A principal of $6000 is invested in an account paying an annual percentage rate of 4%. Find te amount in te account after 5 years if te account is compounded a. semiannually b. quarterly c. montly 8. A principal of $5,000 is invested in an account paying an annual percentage rate of 5%. Find te amount in te account after years if te account is compounded a. semiannually b. quarterly c. montly

6 74 CHAPTER Equations, Inequalities, and Problem Solving MIXED Solve. For Exercises 9 and 30, te solutions ave been started for you. Round all dollar amounts to two decimal places. See Examples 4 and Omaa, Nebraska, is about 90 miles from Lincoln, Nebraska. Irania Scmidt must go to te law library in Lincoln to get a document for te law firm se works for. Find ow long it takes er to drive round-trip if se averages 50 mp. Start te solution: 1. UNDERSTAND te problem. Reread it as many times as needed.. TRANSLATE into an equation. (Fill in te blanks below.) Here, we simply use te formula d = r # t. distance rate or equals # time (round-trip) speed T T T T T = # t Finis wit: 3. SOLVE and 4. INTERPRET 30. It took te Selby family 5 1 ours round-trip to drive from teir ouse to teir beac ouse 154 miles away. Find teir average speed. Start te solution: 1. UNDERSTAND te problem. Reread it as many times as needed.. TRANSLATE into an equation. (Fill in te blanks below.) Here, we simply use te formula d = r # t. distance rate or equals # time (round-trip) speed T T T T T = r # Finis wit: 3. SOLVE and 4. INTERPRET 31. Te day s ig temperature in Poenix, Arizona, was recorded as 104 F. Write 104 F as degrees Celsius. [Use te formula C = 5 1F - 3.] 9 3. Te annual low temperature in Nome, Alaska, was recorded as -15 C. Write -15 C as degrees Fareneit. [Use te formula F = 9 5 C + 3.] 33. A package of floor tiles contains 4 one-foot-square tiles. Find ow many packages sould be bougt to cover a square ballroom floor wose side measures 64 feet. 34. One-foot-square ceiling tiles are sold in packages of 50. Find ow many packages must be bougt for a rectangular ceiling 18 feet by 1 feet. 35. If te area of a triangular kite is 18 square feet and its base is 4 feet, find te eigt of te kite. eigt 36. Bailey, Etan, Avery, Mia, and Madison would like to go to Disneyland in 3 years. Teir total cost sould be $4500. If eac invests $800 in a savings account paying 5.5% interest compounded semiannually, will tey ave enoug in 3 years? 37. A gallon of latex paint can cover 500 square feet. Find ow many gallon containers of paint sould be bougt to paint two coats on eac wall of a rectangular room wose dimensions are 14 feet by 16 feet (assume 8-foot ceilings). 38. A gallon of enamel paint can cover 300 square feet. Find ow many gallon containers of paint sould be bougt to paint tree coats on a wall measuring 1 feet by 8 feet. To prepare for Exercises 43 and 44, use te volume formulas below to solve Exercises Remember, volume is measured in cubic units. 4 ft Cylinder: V = pr Spere: V = 4 3 pr3 39. Te cylinder below as an exact volume of 980p cubic meters. Find its eigt. eigt 7 m 40. Te battery below is in te sape of a cylinder and as an exact volume of 85p cubic millimeters. Find its eigt. r 5 mm eigt 41. Te steel ball below is in te sape of a spere and as a diameter of 1 millimeters. d 1 mm 64 ft 64 ft a. Find te exact volume of te spere. b. Find a -decimal-place approximation for te volume.

7 Section.3 Formulas and Problem Solving Te sperical ball below as a diameter of 18 centimeters. d 18 cm a. Find te exact volume of te ball. b. Find a -decimal-place approximation of te volume. 43. A portion of te external tank of te Space Suttle Endeavour was a liquid ydrogen tank. If te ends of te tank are emisperes, find te volume of te tank. To do so, answer parts a troug c. (Note: Endeavour completed its last mission in 011.) 4. m 4. m 1. m Cylinder Spere a. Find te volume of te cylinder sown. Round to decimal places. b. Find te volume of te spere sown. Round to decimal places. c. Add te results of parts a and b. Tis sum is te approximate volume of te tank. 44. A vitamin is in te sape of a cylinder wit a emispere at eac end, as sown. Use te art in Exercise 43 to elp find te volume of te vitamin. 46. In 1945, Artur C. Clarke, a scientist and science-fiction writer, predicted tat an artificial satellite placed at a eigt of,48 miles directly above te equator would orbit te globe at te same speed wit wic te eart was rotating. Tis belt along te equator is known as te Clarke belt. Use te formula for circumference of a circle and find te lengt of te Clarke belt. (Hint: Recall tat te radius of te eart is approximately 4000 miles. Round to te nearest wole mile.),48 mi 47. Te deepest ole in te ocean floor is beneat te Pacific Ocean and is called Hole 504B. It is located off te coast of Ecuador. Scientists are drilling it to learn more about te eart s istory. Currently, te ole is in te sape of a cylinder wose volume is approximately 3800 cubic feet and wose lengt is 1.3 miles. Find te radius of te ole to te nearest undredt of a foot. (Hint: Make sure te same units of measurement are used.) 48. Te deepest man-made ole is called te Kola Superdeep Boreole. It is approximately 8 miles deep and is located near a small Russian town in te Arctic Circle. If it takes 7.5 ours to remove te drill from te bottom of te ole, find te rate tat te drill can be retrieved in feet per second. Round to te nearest tent. (Hint: Write 8 miles as feet, 7.5 ours as seconds, ten use te formula d = rt.) 49. Earta is te world s largest globe. It is located at te eadquarters of DeLorme, a mapmaking company in Yarmout, Maine. Earta is feet in diameter. Find its exact circumference (distance around) and ten approximate its circumference using 3.14 for p. (Source: DeLorme) 4 mm 15 mm a. Find te volume of te cylinder part. Round to two decimal places. b. Find te volume of te spere formed by joining te two emisperical ends. Round to two decimal places. c. Add te results of parts a and b to find te approximate volume of te vitamin. 45. Amelia Earart was te first woman to fly solo nonstop coast to coast, setting te women s nonstop transcontinental speed record. Se traveled miles in 19 ours 5 minutes. Find te average speed of er fligt in miles per our. (Cange 19 ours 5 minutes into ours and use te formula d = rt.) Round to te nearest tent of a mile per our. 50. Earta is in te sape of a spere. Its radius is about 0.6 feet. Approximate its volume to te nearest cubic foot. (Source: DeLorme) 51. Find ow muc interest $10,000 earns in years in a certificate of deposit paying 8.5% interest compounded quarterly. 5. Find ow long it takes Mark to drive 135 miles on I-10 if e merges onto I-10 at 10 a.m. and drives nonstop wit is cruise control set on 60 mp.

8 76 CHAPTER Equations, Inequalities, and Problem Solving Te calorie count of a serving of food can be computed based on its composition of carboydrate, fat, and protein. Te calorie count C for a serving of food can be computed using te formula C = 4 + 9f + 4p, were is te number of grams of carboydrate contained in te serving, f is te number of grams of fat contained in te serving, and p is te number of grams of protein contained in te serving. 53. Solve tis formula for f, te number of grams of fat contained in a serving of food. 54. Solve tis formula for, te number of grams of carboydrate contained in a serving of food. 55. A serving of casews contains 14 grams of fat, 7 grams of carboydrate, and 6 grams of protein. How many calories are in tis serving of casews? 56. A serving of cocolate candies contains 9 grams of fat, 30 grams of carboydrate, and grams of protein. How many calories are in tis serving of cocolate candies? 57. A serving of raisins contains 130 calories and 31 grams of carboydrate. If raisins are a fat-free food, ow muc protein is provided by tis serving of raisins? 58. A serving of yogurt contains 10 calories, 1 grams of carboydrate, and 5 grams of protein. How muc fat is provided by tis serving of yogurt? Round to te nearest tent of a gram. REVIEW AND PREVIEW Determine wic numbers in te set 5-3, -, -1, 0, 1,, 36 are solutions of eac inequality. See Sections 1.4 and x x x x - 3 Ú In your own words, explain wat real numbers are solutions of x In your own words, explain wat real numbers are solutions of x 7 1. CONCEPT EXTENSIONS Solar system distances are so great tat units oter tan miles or kilometers are often used. For example, te astronomical unit (AU) is te average distance between Eart and te sun, or 9,900,000 miles. Use tis information to convert eac planet s distance in miles from te sun to astronomical units. Round to tree decimal places. Te planet Mercury s AU from te sun as been completed for you. (Source: National Space Science Data Center) Planet Miles from te Sun AU from te Sun Mercury 36 million Venus 67. million Eart 9.9 million Mars million Jupiter million Saturn million Planet Miles from te Sun AU from te Sun Uranus 1783 million Neptune 793 million Pluto (dwarf planet) 3670 million 73. To borrow money at a rate of r, wic loan plan sould you coose one compounding 4 times a year or 1 times a year? Explain your coice. 74. If you are investing money in a savings account paying a rate of r, wic account sould you coose an account compounded 4 times a year or 1 times a year? Explain your coice. 75. To solve te formula W = g - 3gt for g, explain wy it is a good idea to factor g first from te terms on te rigt side of te equation. Ten perform tis step and solve for g. 76. An orbit suc as Clarke s belt in Exercise 46 is called a geostationary orbit. In your own words, wy do you tink tat communications satellites are placed in geostationary orbits? Te measure of te cance or likeliood of an event occurring is its probability. A formula basic to te study of probability is te formula for te probability of an event wen all te outcomes are equally likely. Tis formula is Probability of an event = number of ways tat te event can occur number of possible outcomes For example, to find te probability tat a single spin on te spinner below will result in red, notice first tat te spinner is divided into 8 parts, so tere are 8 possible outcomes. Next, notice tat tere is only one sector of te spinner colored red, so te number of ways tat te spinner can land on red is 1. Ten tis probability denoted by P(red) is P(red) Ω Find eac probability in simplest form. 77. P(green) 78. P(yellow) 79. P(black) 80. P(blue) 81. P(green or blue) 8. P(black or yellow) 83. P(red, green, or black) 84. P(yellow, blue, or black) 85. P(wite) 86. P(red, yellow, green, blue, or black) 87. From te previous probability formula, wat do you tink is always te probability of an event tat is impossible to occur? 88. Wat do you tink is always te probability of an event tat is sure to occur?