Algebra Solutions - Speech or Presentation Example

Algebra Exercise 2: 122. Diversification: Helen has 5) of her portfolio in US stocks 8) of her portfolio in European stocks, and 10) of her portfolio in Japanese stocks. The remainder is invested in municipal bonds. What fraction of her portfolio is invested in municipal bonds? What percent is invested in municipal bonds? Fraction invested in Municipal bonds = 1 – [(1/5) + (1/8) + (1/10)] = 1 – (17/40) = 23/40 Percent invested in Municipal bonds = (23/40) * 100 = 57.5% Exercise 1.3: 106. Net Worth: Melanie’s house is worth $125,000, but she still owes $78,422 on her mortgage. She has $21,236 in a savings account and has $9477 in credit card debt. She owes $6131 to the credit union and figures that her cars and other household items are worth a total of $15,000. What is Melanie’s net worth? Net worth = 125000 – 78422 + 21236 – 9477 – 6131 + 15000 = $67,206 Exercise 1.5: 130. Population of Mexico: In 2006, the population of Mexico was 107.4 million. If Mexico’s population continues to grow at an annual rate of 1.43%, then the population in 2020 will be (107.4) (1.0143)14 million. a. Find the predicted population in 2020 to the nearest tenth of a million people. Population of Mexico in 2020 = 107.4 (1.0143)14 = 131.0 million b. Use the result of exercise 129 to determine whether the United States or Mexico will have the greater increase in population between 2006 and 2020. Population of US in 2020 = 300.2 (1.0105)14 = 347.5 million Increase in US population = 347.5 – 300.2 = 47.3 million Increase in Mexico population = 131 – 107.4 = 23.6 million US will have the greater increase in population between 2006 and 2020. Exercise 1.6: 100. Forensics: A forensic scientist uses the expression 72.6 + 2.5T to estimate the height in centimetres of a female with a tibia of length T cms. If a female skeleton has a tibia of length 32.4 cm, then what was the height of the person? Find the length of your tibia in cms, and use the expression in this exercise to estimate your height. Height = 72.6 + 2.5T = 72.6 + 2.5 (32.4) = 72.6 + 81 = 153.6 cm Tibia Length = 41.4 cm Height = 72.6 + 2.5 (41.4) = 72.6 + 103.5 = 176.1 cm 104. Crop Circles: The expression r2 gives the area of a circle with radius r. How many square meters of wheat were destroyed when an alien ship made a crop circle of diameter 25 m in the wheat field at the Southwind Ranch? Round to the nearest 10th. Find  on your calculator. Radius of the ship = (25/2) = 12.5 m Area of the Ship = r2 = (12.5)2 = 490.0 sq. meters Exercise 1.8: 124. Marriage penalty eliminated: The value of the expression 4220 + 0.25 (x – 30,650) is the 2006 federal income tax for a single tax payer with a taxable income of x dollars, where x is over $30,650 but not over $74,200. a. Simplify the expression. 4220 + 0.25 (x – 30,650) = 4220 + 0.25x – 7662.50 = 0.25x – 3442.50 b. Find the amount of tax for a single tax payer with taxable income of $40,000. When x = 40,000 Tax = 0.25 (40,000) – 3442.50 = 10,000 – 3442.50 = 6557.50 c. Who pays more, a married couple with a joint taxable income of $80,000 or two single tax payers with taxable incomes of $40,000 each? 1. Married Couple Tax = 8440 + 0.25 (x – 61,300) = 8440 + 0.25 (80,000 – 61,300) = 8440 + 4675 = 13,115 2. Tax for 2 singles = 2 * [0.25(40,000) – 3442.50] = 2 * (6557.50) = 13,115 Exercise 2.1: 92. (7r/12) = -14 7r = -168 r = -24 94. (1/3)s + (7/9) = (4/3)s (1/3)s – (4/3)s = -7/9 -s = -7/9 s = 7/9 96. World Grain Demand: Freeport McMoRan projects that in 2010 world grain supply will be 1.8 trillion metric tonnes and the supply will be only (3/4) of the world grain demand. What will world grain demand be in 2010? Supply = 1.8 trillion metric tonnes World Grain demand = x ¾ x = 1.8 trillion metric tonnes x = 2.4 trillion metric tonnes Exercise 2.2: 78. 0.3x = 1 – 0.7x 0.3x + 0.7x = 1 1x = 1 x = 1 86. 9 – (2m/7) = 19 -2m/17 = 10 -2m = 170 m = -85 90. 3 (x – 0.87) – 2x = 4.98 3x – 2.61 – 2x = 4.98 x – 2.61 = 4.98 x = 7.59 94. Fahrenheit Temperature: Water boils at 212 F. a. Determine the Celsius temperature at which water boils. From the graph, water boils at 100 C. b. Find the Fahrenheit temperature of hot tap water at 70 C by solving the equation 70 = (5/9) (F – 32) (5/9) (F – 32) = 70 (5/9)F – (160/9) = 70 (5/9) F = 70 + (160/9) (5/9) F = (790/9) 5F = 790 F = 158 96. Perimeter of a Triangle: The perimeter of the triangle is 12 m. Determine the values of x, x+1 and x+2 by solving the equation x + (x+1) + (x+2) = 12 3x + 3 = 12 3x = 9 x = 3 Hence the three values are 3, 4 and 5. Exercise 2.3: 14. (x/2) – (x/3) = 5 Multiply both sides by 6, 6 * [(x/2) – (x/3)] = 6 * 5 3x – 2x = 30 x = 30 22. (1/15)k + 5 = (1/6)k – 10 Multiply both sides by 30, 30[(1/15)k+5] = 30[(1/6)k – 10] 2k + 150 = 5k – 300 3k = 450 k = 150 32. 0.5b + 3.4 = 0.2b +12.4 10(0.5b + 3.4) = 10(0.2b + 12.4) 5b + 34 = 2b + 124 5b – 2b = 124 – 34 3b = 90 b = 30 54. -6x + 3 = -7 – 5x -6x + 5x = -7 - 3 -x = -10 x = 10 68. 3 – 3(5 – x) = 0 3 – 15 + 3x = 0 (Add 12 to each side) 3x = 12 (Divide both sides by 3) x = 4 80. -3 – 4 (t – 5) = -2 (t + 3) + 11 -3 -4t + 20 = -2t – 6 + 11 -2t = -12 t = 6 88. 0.08x + 0.5 (x + 100) = 73.2 0.08x + 0.5x + 50 = 73.2 0.58x = 23.2 x = 40 94. Raising Rabbits: Before Roland sold two female rabbits, half of his rabbits were female. After the sale, only one third of his rabbits were female. If x represents the original number of rabbits, then (1/2) x – 2 = (1/3) (x – 2) Solve this equation to find the number of rabbits that he had before the sale. (x/2) – (x/3) = 2 – (2/3) (x/6) = 4/3 x = 8 Exercise 2.4: 32. Solve for x. x – a = -x + a + 4b x – a = -x + a + 4b x + x = a + a + 4b 2x = 2a + 4b x = a + 2b 54. Solve for y. y + (1/2) = -(1/3) (x + ½) y + (1/2) = -(1/3) (x + ½) y = -x/3 – 1/6 y = - (2x + 1) / 6 72. S = n (n + 1) (2n + 1) /6 n S 1 1 2 5 3 14 4 30 5 55 82. Finding MSRP. What was the MSRP for the Hummer H1 that was sold for $107,272 after an 8% discount? x – 0.08x = 107272 0.92x = 107272 x = 116600 Hence the MSRP is $116,600. 98. Cowling’s Rule: Cowling’s rule is another method for determining the dosage of a drug to prescribe to a child. For this rule, the formula, d = D (a + 1) / 24 gives the child’s dosage d, where D is the adult dosage and a is the age of the child in years. If adult dosage of a drug is 600 mgs and a doctor uses this formula to determine that a child’s dosage is 200 mgs, then how old is the child? When D = 600 and d = 200, 600 (a +1) / 24 = 200 600 (a + 1) = 4800 a + 1 = 8 a = 7 Hence the child is 7 years old. Exercise 2.6: 8. Find two consecutive odd integers whose sum is 56. x + (x+2) = 56 2x + 2 = 56 2x = 54 x = 27 Hence the two consecutive odd integers are 27 and 29. 14. Consecutive even integers: Find 4 consecutive even integers whose sum is 340. x + (x+2) + (x+4) + (x+6) = 340 4x + 12 = 340 4x = 328 x = 82 Hence the four numbers are 82, 84, 86 and 88. Exercise 2.7: 10. Chrysler Sebring: After getting a 15% discount on the price of a new Chrysler Sebring Convertible, Helen paid $27,000. What was the original price of the convertible to the nearest dollar? Original Price = x x – 0.15x = 27000 0.85x = 27000 x = 31764.70 Hence the original price was $31,764.70. 14. Toyota Corolla: Gwen bought a new Toyota Corolla. The selling price plus the 8% state sales tax was $15,714. What was the selling price? Selling Price = x x + 0.08x = 15,714 1.08x = 15714 x = $14,550 Hence the selling price is $14,550. 18. High Risk Funds: Of the $50,000 that Natasha pocketed on her last real estate deal, $20,000 went to charity. She invested part of the remainder in Dreyfus New Leaders Fund with an annual yield of 16% and the rest in Templeton Growth Fund with an annual yield of 25%. If she made $6,060 on these investments in 1 year, then how much did she invest in each fund? Total Investment = 50000 – 20000 = 30000 Investment in Dreyfus = D Investment in Templeton = T Hence, D + T = 30000  Eqn. 1 As Natasha has made 6060 on these investments, (D * 0.16) + (T * 0.25) = 6060 0.16D + 0.25T = 6060  Eqn. 2 Eqn. 1 * 0.16  0.16D + 0.16T = 4800 Eqn. 2  0.16D + 0.25T = 6060 Subtracting  -0.09T = -1260 T = 14000 D = 30000 – 14000 = 160000 Hence Natasha invested $16,000 in Dreyfus New Leaders Fund and $14,000 in Templeton Growth Fund. Exercise 2.8: 88. Building a Ski-ramp: Write an inequality in the variable x, for the degree measure of the smallest angle of the triangle shown in the figure, given that the degree measure of the smallest angle is atmost 30 degrees. The smallest angle = 180 - x – (x +8) = 172 – 2x 172 – 2x < 30 Exercise 2.9: 68. Car Selling: Ronald wants to sell his car through a broker who charges a commission of 10% of the selling price. Ronald still owes $11,025 on the car. Ronald must get enough to atleast pay for the loan. What is the range of the selling price? Selling Price = x Commission = 0.1x Cash to Ronald = x – 0.1x = 0.9x 0.9x has to be atleast equal to 11,025 for Ronald to pay for the loan. 0.9x > 11,025 (Dividing by 0.9) x > 12,250 Hence the selling price of the car has to be atleast $12,250. 74. Different Weights: Professor William counts his midterm as (2/3) of the grade and his final as (1/3) of the grade. Wendy scored only 48 on the midterm. What range of scores on the final exam would put Wendy’s average between 70 and 79 inclusive? Final Exam Score = x Average Score = (48*2/3) + (x/3) = 32 + (x/3) Hence, in order to get a score between 70 and 79, 70 < 32 + (x/3) < 79 38 < (x/3) < 47 (Subtracting 32) 114 < x < 141 (Multiplying by 3) Hence, Wendy has to score 114 to 141 inclusive to get an average score between 70 and 79 inclusive. Warm ups 2.1: 1. True 2. True 3. False. We have to multiply each side by (4/3). 4. True. 5. True. 6. True. 7. True. 8. True. 9. True. 10. True. Warm ups 2.2: 1. True. 2. True. 3. True. 4. False. Subtract 5 from each side and then subtract 7x from each side. 5. True. 6. False. First add 7 to each side and then divide by 3. 7. True. 8. True. 9. True. 10. True. Warm ups 2.3: 1. True. 2. True. 3. False. The equation is equivalent to 20x + 3x = 800. 4. False. The Solution is {-8/3} 5. False. The equation has a solution. 6. False. The equation is inconsistent. 7. True. 8. True. 9. True. 10. True. 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