Solutions to Practice Problems in the Text

Transcription

1 Solutions to Practice Problems in the Text Chapter One: Fundamentals of Mathematical Modeling Practice Set 1-1. d = rt 1 = 0t [Divide both sides by 0.] t =. hours. d = rt 170 = r(.) [Divide both sides by..] 68 mph = r. I = Prt I = ($000)(0.0)() = $00. I = Prt 616 = (800)r() 616 = 1100r 0.0 = r. A = r A = = (.1)() = 8.6 in 6. A 1.6 = (.1)r = r = ft = r 7. A = ½bh 6 = ½(8)h 6 = h in = h 8. A = ½bh 10 = ½b(0) 10 = 0b 7. cm = b. A = s( s a)( s b)( s c) 1(1 )(1 1)(1 1) 1(10)()() 00 0 in 10. A = s( s a)( s b)( s c) 6(6 )(6 )(6 ) 6()()(1) 6 6 in INSTRUCTOR

2 o 11. C = F C o 1. C = F 6 0 C o 1. F C F o 1. F C F 1. m = y x y x 1 1 ( ) y y m = 1 x x ( 1) z = x x s x x z = 1 s a + b = c + = c + 16 = c = c = = c 0. a + b = c 1 + = c 1 + = c 16 = c INSTRUCTOR

3 16 = 1 = c nt r M = P $78. 6 n 1 1 1*10 10 nt r M = P1 1, ,000 1 $1,11. 8 n 1 1 1* 60 b b ac () (1)( 6). x = a (1) 7 1 or b b ac ( ) ( ) ()( ) x = a () or y = Ae rn = 1,00,000e 0.0*7 = 1,00,000e 0.8 =,0,1 bacteria 6. y = Ae rn =,000e 0.01*10 =,000e 0.1 =,8. =,8 people 7. I =Prt I Pt I Pt Prt [Divide both sides by Pt.] Pt r 8. V = lwh V lwh [Divide both sides by lh.] lh lh INSTRUCTOR

4 V lh w 1. A = bh A = bh [Multiply both sides by.] A h b [Divide both sides by h to solve for b.] 1 0. V = Bh V = Bh [Multiply both sides by.] V B h [Divide both sides by B to solve for h.] 1. P = L + W P W = L+W W P W = L P W L [Subtract W from both sides.] [Divide both sides by.]. P = L + W P L = L+W L P L = W P L W [Subtract L from both sides.] [Divide both sides by.]. A = ½(B + b)h (A) = [½(B + b)h] [Multiply both sides by.] A = (B + b)h A = Bh + bh [Use the Distributive Property.] A bh = Bh + bh bh [Subtract bh from both sides.] INSTRUCTOR 6

5 A bh = Bh A bh B [Divide both sides by h to solve for B.] h. A = ½(B + b)h (A) = [½(B + b)h] [Multiply both sides by.] A = (B + b)h A h [Divide both sides by (B + b) to solve for h.] B b. x + y = 6 x + y x = 6 x y = 6 x y = - [Subtract x from both sides.] [Divide both sides by to solve for y.] 6. x y = 10 x y x = 10 x [Subtract x from both sides.] -y = 10 x y = x [Divide both sides by -1.] 7. A = x y A = x + y [Multiply both sides by.] A y = x + y - y [Subtract y from both sides.] A y = x 8. A = x y A = x + y [Multiply both sides by.] INSTRUCTOR 7

6 A x = x + y - x [Subtract x from both sides.] A x = y. F C F = C [Multiply through by.] F 160 = C [Subtract 160 from both sides.] F 160 = C 160 F C [Divide both sides by.] F = C [Factor out.] 0. C = F C + C + C = 160 F [Use the distributive property.] = F = F C F [Add 160 to both sides.] [Multiply through by.] 1. W 10 BMI = , Normal H 7. W 160 BMI = , Obese H 60 Practice Set 1- min min 1. hr 10 min 8 INSTRUCTOR 8

7 . 1 hr min 60 min min in in ft 8 in 60 ft in 6 in 60 in ft yd 6 ft ft 6. yd 1 ft 6 1 ft ft weeks 6 days 16 days 16 days 7 8. days 1 week days 7 days 7. 1 ml L 1000 ml ml ml L 00 ml 000 ml 6 0miles 11. mi./gal.. gal 0 miles 1. 0 gal. mi./gal. $.0 1. $0./min. 10min $ kwh $0.1/kWh $8 1. $.80/day 10 days $0 16. $8.0 / hr 0 hr 17. lb = 1.6 lb./person 1 people INSTRUCTOR

8 .lb lb./person 1 people 1. x x = ()() x = 1 x = 1 [cross-multiplication property] 0. x 6 ()() = (x)(6) [cross-multiplication property] 6 = 1x = x x (x)(0) = (16) 0x = 60 x = 7 x (x)(1) = (7)(8) 8x = 6 x = [cross-multiplication property] [cross-multiplication property] x 6 x 18. (x + 6) = (x - 18) [cross-multiplication property] 1x + 0 = 70x - 60 [distributive property] 1x x = 70x x -x + 0 = -60 -x = x = -660 x = 1. x x 7 8 8(x ) = 7(x + ) [cross-multiplication property] 8x 6 = 1x + 1 [distributive property] 8x 6 1x = 1x + 1 1x 1x 6 = 1 1x = INSTRUCTOR 0

9 1x = 70 x =. 1 x 1 18 x (1)(x) = 18(x - 1) 1x = 18x x - 18x = 18x x -x = -18 x = 6 [cross-multiplication property] [distributive property] 6. x x (x + ) = (x + ) [cross-multiplication property] 6x + = x + 0 [distributive property] 6x + x = x + 0 x x + = 0 x + = 0 x = x 1 x (x - ) = 18(x + 1) [cross-multiplication property] 7x - = 18x + 18 [distributive property] 7x x = 18x x x - = 18 x - + = 18 + x = 7 x = x 8 x (x + 8) = 6(x 8) [cross-multiplication property] x + = 1x 8 [distributive property] x + -1x = 1x 8 1x -x + = -8 -x + = -8 -x = -7 x = 8. Unit rate equals cost divided by the number of square feet. $ 100 ft = $1./ft 0. Unit rate equals cost divided by the number of yards. $7.80 yd = $11./yd 1. Cost per ounce = $8. 16 oz = $ per ounce INSTRUCTOR 1

10 . Cost per can = $7.1 6 cans = $1.1 per can. Brand X: $1. 8 oz = $0.186/ ounce Brand Z: $.1 1 oz = $ /ounce Brand Z is the better buy because it costs less per ounce.. five -L colas: $. = $0./bottle three -L colas: $. = $ /bottle Purchasing five -L colas for $. is a better buy.. Let 10 cal x. cup 1 cup (1)(10) = ¾x [Cross multiply.] 10 = ¾x 160 calories = x [Multiply both sides by /.] 6. Let 8 oz 8 oz. 0 cal x 8x = (0)(8) 8x = 0 x = 1 calories [Cross multiply.].dozen x dozen 7. Let. 1.cups cups ()(.) = 1.x [Cross multiply.] 7. = 1.x 6 = x Answer is 6 dozen or 7 muffins. 8. If each of 0 students has cookies, then 60 cookies are needed = doz. dozen dozen Let.cups x x = ()(.) [Cross multiply.] x = 1. x = = 1 cups 6. Let ft in x. Convert the measurements to inches as follows: ft in = (1) 10. ft 0 ft + = 6 inches; 10. ft = in = 16 inches; 0 ft = 0 1 in = 0 inches. Now substitute: 6 in x 16 in 0 in (6)(0) = 16x INSTRUCTOR

11 160 = 16x 11. inches = x Converting to feet and inches: 11. inches 1in/ft = 10 ft. 1. in. ft 10in x 0. Let. Convert the measurements to inches as follows: ft 10 in = (1) 8 ft in 1 ft + 10 = 70 inches, 8 ft in = 8(1) + = 10 inches, and 1 ft = 1(1) = 168 inches. Now substitute: 70 in x. 10 in 168 in (70)(168) = 10x [Cross multiply.] = 10x 11 inches = x (11 inches = feet inches) 1.8 A. A 1. Let. 18V x (18)(.) = 1.8A [Cross multiply.] 7. = 1.8A V = A 10 A A. Let 0 V x 10x = ()(0) 10x = 10 x = 1 V [Cross multiply.]. Let. Let 110lb 00 lb 1. lb x 110x = (1.)(00) [Cross multiply.] 110x = 880 x =.77 or approximately. lb. 110lb 1 lb 1. lb x 110x = (1.)(1) [Cross multiply.] 110x = 007 x = or approximately 7. lb. 10lb x. Let 00 ft 00 ft (10)(00) = 00x [Cross multiply.] 000 = 00x INSTRUCTOR

12 1. lb = x 6. Let 0 6 oz [ gallons = 6 oz.] 1 x 0x = (6)(1) [Cross multiply.] 0x = 6 x = 1.8 oz. 7. Let $0 x. 100 words 100words (0)(100) = 100x [Cross multiply.] 60,000 = 100x $00 = x 8. Let. Let 100 doughnuts 00 doughnuts hours x 100x = ()(00) 100x = 700 x = 1.8 hr. 1 adult adults 1 children x x = ()(1) x = children [Cross multiply.] [Cross multiply.] 1 adult x 0. Let 6 infants 1 infants 1 = 6x [Cross multiply.]. adults = x Therefore, adults are needed for 1 infants. 1. Let 1inch. 7 inches 8 feet x x = (8)(.7) x = feet [Cross multiply.] 1inch 1. 7 inches Let 8 feet x x = (8)(1.7) x = 1. feet Dimensions are ft 1. ft [Cross multiply.] INSTRUCTOR

13 . Let 1inch x 10 miles 18 miles (1)(18) = 10x [Cross multiply.] 18. in = x. Let Mach1 Mach mph x 1x = (.1)(761.) x =.7 mph [Cross multiply.]. Let Mach1 x 761. mph 10 mph 761.x = (1)(10) [Cross multiply.] x =. (Mach.) 00 ft 1000 ft. Let 1 day x (1)(1000) = 00x [Cross multiply.] 1000 = 00x. days = x 6. Let 8 sq / day 1 roofer x roofers ()(8 sq/day) = x [Cross multiply.] 16 sq/day = x Since roofers can do 16 squares per day, they can do squares in days. 7. Let 0 mg 60 mg. 1mL x 0x = (1)(60) 0x = 60 x = 1. ml [Cross multiply.] 8. Let 100 mg x. 1kg. kg (100)(.) = 1x [Cross multiply.] 0 mg = x. Let 1.8 mi x 0 min min (1.8)() = 0x [Cross multiply.] 81 = 0x.7 miles = x INSTRUCTOR

14 60. Let 1mph x 1.60 kph 8 kph (1)(8) = 1.60x 8 = 1.60x 60. mph = x Practice Set x + 6. x -. 7x. + x. ( + x) 6. (x 6) 1 7. x 8. x x. Let Bill s salary = x. Then, Ann's salary = x + $ Let Monday s attendance = x. Then, Friday s is ½x Let the width = x. Then, the length = x Let the short rod = x. Then, each subsequent rod is 1 inch longer: x + 1 and x x + = x 10 x + x = x 10 x x + = -10 x + = -10 x = -1 x = x 16 = 80 8x = x = 6 x = 1 1. Emily has saved x. Elena has saved x (twice as much). x + x = 7 x = 7 x = so Emily has saved $ and Elena has saved ($) = $8. INSTRUCTOR 6

15 16. Let the lower grade = x. The higher grade is x + 6. x + (x + 6) = 18 x + 6 = 18 x = 18-6 x = 7 x = 6 so the lower grade is 6 and the higher is =. 17. Let x = the shorter piece. The longer piece will be x. x + x = 60 meters x = 60 meters x = 0 meters The shorter piece is 0 meters long and the longer piece is (0) = 0 meters long. 18. Let Joshua s age = x and Mario s age be x x + (x + 18) = 6 x + 18 = 6 x = 6 18 x = 8 x = 1 So, Joshua is 1 years old and Mario is = years old. 1. Using the definition of an average, let x equal the missing grade x x = (0) 7 + x = x = 60-7 x = 86 In order to have a 0 average, she must make 86 on the last test. 0. Using the definition of an average, let x equal the missing grade x x = (8) 1 + x = x = 0-1 x = 101 In order to have a B average, she must make 101 on the last test. On a normal grading scale, this will be impossible. 1. Let x = the number of kilowatt-hours. Write the equation and solve. $ $0.1x = $ x = x = 6.78 INSTRUCTOR 7

16 x = kwh (rounded) 0.1. Let x = number of out of area minutes. $. + $. + $0.0x = $ x = x = x = 0 x = 7 She was charged for 7 out of area minutes.. Let x = the taxi fare. Since 1/11 of a mile costs $0.0, the cost of one mile = $ = $.0. x = $ ($.0/mile) + $1.0 = $.00. Let x = the taxi fare. Since 1/ of a mile costs $0.0, the cost of one mile = $0.0 = $.00. x = $ ($.00/mile) = $.0. The opponent scored x points in the game. Mighty Mites scored points. This amount () is x - 1. x - 1 = x = + 1 x = 0 x = 0 The opponent scored 0 points. 6. Let the number of games Jim won = x and the number that Gaylord won = x +. x + (x + ) = x + = x + = x = 0 x = 1 Jim won 1 games and Gaylord won 1 + = 1 games. 7. Let the integers be x, x + 1 and x +. Write the equation for the sum and solve. x + (x + 1) + (x + ) = 87 x + = 87 x + = 87 x = 8 x = 8 The integers are x = 8, x + 1 = and x + = Let the integers be x, x + 1 and x +. Write the equation for the sum and solve. x + (x + 1) + (x + ) = 100 INSTRUCTOR 8

17 x + = 100 x + = 100 x = 7 x =... Since this number is not an integer, there is not a set of consecutive integers that satisfies the requirements of this problem.. Let the odd integers be x, x +, and x +. Write the equation for the sum and solve. x + (x + ) + (x + ) = -7 x + 6 = -7 x = -7 6 x = -7 x = - The three integers are x = -, x + = -1, and x + = Let the odd integers be x, x +, and x +. Write the equation for the sum and solve. x + (x + ) + (x + ) = 10 x + 6 = 10 x = 10 6 x = 17 x = The three integers are x =, x + = 01, and x + = Let x = the value of the lot. Then, 6.x = the value of the house. x + 6.x = $17,000 7.x = $17,000 x = $,. So, the lot is worth approximately $, and the house is worth about $11,667.. Let x = the value of the lot. Then, 7x = the value of the house. x + 7x = $16,000 8x = $16,000 x = $0,00 So, the lot is worth approximately $,00 and the house is worth about $1,00.. x =. x = yen = $ yen / dollar 6 pounds = $ pounds / dollar distance 00mi. time. 6881hr = about hr 6 min speed 170.6mi / hr INSTRUCTOR

18 6. x = 00 miles 7.60 mph = 6.70 hours. Convert 0.70 to minutes by multiplying 0.70(60) =.1 to give the answer of approximately hours and minutes. 7. Let the number of males and females at the beginning of the semester = x. If 8 males drop the class, there are x 8 males remaining. Now the number of females is twice the number of males remaining. x = (x 8) x = x 16 x x = x 16 x -x = -16 x = 16 There were 16 males and 16 females at the beginning of the semester. 8. Let x = the number of Democrats and the number of Republicans in the Senate. x + x + = 100 x + = 100 x + = 100 x = 8 x = There were Democrats and Republicans in the Senate.. Let x = the number of bags of apples. x + x = 7x = x = 6 There are 6 bags containing lb. of apples and 6 bags containing lb. of apples. 0. Let x = the number of seats in Theater B, x + 10 = the number of seats in Theater A. (x + 10) + x + 70 = 800 x + 0 = 800 x = x = 80 x = 10 The number of seats in Theater A is x + 10 = = Let x = profit from the automotive division. Then x 7 represents the profit from financial services. x + (x 7 million) = 8 million x 7 million = 8 million x 7 million + 7 million = 8 million + 7 million x = 76 million x = 78 million The profit from the automotive division was 78 million and from financial services INSTRUCTOR 0

19 was 78 million 7 million = 10 million.. Let x = salary of Palmisano and x - $.1 million = Moonvie s salary. x + x -.1 million = 6.7 million x -.1 million +.1 million = 6.7 million +.1 million x = 60.6 million x = 0. million Therefore, Palmisano made $0. million and Moonvie made $6.1 million in Let x = attendance at the Ohio State game and x 10 = attendance at the Penn State game. Total attendance for the two teams in 010 was 0,1. x + x - 10 = 0,1 x - 10 = 0,1 x = 0, x = 10,6 x = 10,78 Therefore, the attendance at the Ohio State game was 10,78 and the attendance at the Penn State game was 10,78 10 = 10,.. Let x = the middle-sized piece. The longest is x and the shortest is x -. The total length of the pipe is 0 inches. Therefore, x + x + x - = 0 7x - = 0 7x - + = 0 + 7x = 6 x = Therefore, the middle sized piece is in., the longest is () = 7 in. and the shortest is x - = () - = in.. Let x = the original price of the radio. Savings = 1% of retail price or 0.1x. x 0.1x = $ x = $17.6 x = $1. The original price of the radio was $ Let x = the original price of the watch. Savings = % of retail price or 0.x. x 0.x = $ x = $168.7 x = $ The original price of the watch was $ Let x = the wholesale price of the shoes. The markup amount is 6% times x. x + 0.6x = $ x = $1.0 x = $76 The wholesale price of the shoes was $ INSTRUCTOR 1

20 8. Let x = the wholesale price of the shoes. The markup amount is 0% times x. x + 0.0x = $6 1.x = $6 x = $ The wholesale price of the shoes was $.00.. Let x = Drema s contributions. The company s contributions = 0% of x. x + 0.0x = $100 1.x = $100 x = $1000 Drema deposited $1000 into the account. 0. Let x = the population four years ago. The growth of the population is % of x. x + 0.0x = x = 88 x = 80 There were 80 people in the town four years ago. W BMI = , Normal H 60 W 1.. BMI = , Obese H 67 Chapter 1 Review Problems 1.. I = Prt I t [Divide both sides by Pr.] Pr. x + y = y = -x + [Subtract x from both sides.] y x [Divide all terms by.]. C d. a + b + c = P c = P a b [Subtract a and b from both sides.]. 7 min 7 min hr 180 min 0 INSTRUCTOR

21 6. weeks 8 days 1 days 1 days 7. 6 in ft 6 60 in in 1 10 $ hr $1/hour bushels. 8 trees. bushels/tree $ lb $.66/lb x 7 7x = 1 1 x = 7 8 x x = x = 6 x 8 x - 1 = x = + 1 x = 6 x = x x 1 7 1x - = 1x - 7 1x - - 1x = 1x - 7-1x -x - + = x = x = A =½bh. Substitute: A =½( in)( in) = 6 in INSTRUCTOR

22 00 L x 16. Let 0 min 0 min 7000 = 0x 10 liters = x. Cross multiply. x 17. Let. Cross multiply. 76 = x 6 dentists = x 18. Let x = total number of minutes that you talk. Then, $1. + (x - )($0.8) = $ x = x = x = x =.6 x = 1 The call was 1 minutes long. 1. Let x = number of minutes per month. Then, $0 + $0.6x = $ x = x =.68 x = 18 minutes 0. Let x = labor time (per hour). Then, $0 + $0x = $ x - 0 = x = 7 x =. hours 1. Let the shorter piece be x and the longer be x +. Then, x + (x + ) = x + = x + - = - x = 0 x = 1 in (short piece) and x + = 1 + = 18 in (long piece). Let the short piece = x, the middle-sized piece = x + 1, and the long piece = x. Then, x + (x + 1) + x = 7 x + 1 = 7 x = 7-1 x = 6 x = 1 cm (short) x + 1 = (1) + 1 = 7 cm (middle) x = (1) = cm (long). Let the calculator be x and the cassette player be x Then, x + (x + 10) = 08 x + 10 = 08 INSTRUCTOR

23 x = x = 68 x = $.00 which is the cost of the calculator. Let n = the number of nickels and n - = the number of dimes. Then, n + (n - ) =. n - = n - + = + n = n = 18 There are 18 nickels and dimes in the bank for a total of $.0.. Use the formula d = rt and substitute values. 18 mi = ( mph)(t) t =. hours 6. T= UN + F Substitute the given values and solve. $16,70 = $1N + $ = 1N = 1N N = 0 units produced 6. First remove the parentheses by use of the distributive property, then add to the 1. + (x + ) = + 6x + 1 = 6x x + = 6x x + -x = 6x x = x. for any triangle, A + B + C = 180 Let A = the first angle The second angle, B = A The third angle, C = A + A + A = 180 6A = 180 A = 0, B = (0 0. First convert all the measurements to the same units, like inches. ft in = 6 in, 10 ft 6 in = 16 in, and ft = 6 in Now set up a proportion like: 6in x 16in 6in Cross multiply: (6in)(6in) = (16in)(x) Divide by 16in: x = 1in = 6ft INSTRUCTOR

24 Chapter 1 Test 1. V = lwh V w [Divide both sides by lh.] lh. h = vt - 16t h + 16t = vt h 16t v t [Add 16t to both sides.] [Divide both sides by t.]. hr hr 1 day hr ft 10 in 160 in 160 in $1.0 days. $10.0/ day 7. lb lb / person 6 people 0 eggs 7. eggs / chicken 10 chickens x = 6x 70 x 6 18 x x 8 (x - ) = 8(x + ) x - 6 = 16x + x x = 16x x -7x - 6 = -7x = + 6-7x = 60 x = 60 7 INSTRUCTOR 6

25 10. Let x = defective bulbs. Then, 8 x. 10 8x = 10 x = 18 bulbs 11. Let x = parts produced. Then, 00 parts x. 0 min min 0x = 100 x = 67 parts 1. Given P = L + W. Substitute the given values into the formula. P = (0 ft.) + (1 ft.) = 6 ft. 1. Company A Plan: $ m (where m = miles driven) Company B Plan: $ m To find the number of miles where the two costs are equal, set these two expressions equal. $ m = $ m m m = m m 0 = m 0-10 = m = 0.0m 0 = m, so at 0 miles the costs are the same for both plans 1. Let the number of passengers on one ship = x. The second ship holds twice as many passengers = x Write the equation x + x = 0 and solve. x = 0 x = 70 so the smaller ship holds 70 passengers 1. Let Sarah's age be x. Then, Michelle's age is x Write the equation and solve. x + x - 10 = 6x - 10 = 6x = x = x = so Sarah is years and Michelle is ()-10 = years. 16. Let the short board = x and the longer one be x + 1. Write the equation. x + x + 1 = 1 x + 1 = 1 x = 1-1 x = 0 x = so the short board is ft. and the longer is () + 1 = 16 ft. 17. x + 1.% of x is now $.8 per gallon x + 0.1x = $.8 INSTRUCTOR 7

26 1.1x = $.8 x = $. per gallon 18. P % of P is now 8,000 people, after 8 years P 0.P = 8, P = 8,000 P =,87. =,88 whole people 8 years ago Now divide the amount of decrease in population by 8 years to get a rough average yearly decrease in population:,88 8,000 = 788 people in 8 years 788/8 = 87. or about 88 people per year decrease in population 1. The average of the four grades needs to equal at least 8 so: ( x )/ = 8 [multiply by ] x = x = 6 x = is the least the student can make an have an overall average of 8 0. Let L = lot value, H = house value = 7.L L + 7.L = $1,000 8.L = $1,000 L = $17,88. or about $17,88 for the lot H = 7.L = $1, or about $18,118 for the house INSTRUCTOR 8