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Solutions to End-of-Chapter Questions and Problems 529 10. Assuming the property is rectangular, the total area is 300 440 132,000 square yards. The owner wants to keep a piece that measures 110 44 4,840 square yards. Therefore, the area he wants to sell is 132,000 4,840 127,160 square yards. 11. Convert all lengths to inches. Then consider the outside rectangle that measures 207 by 216 inches. Its area is 207 216 44,712 square inches. From this, subtract two rectangles, one measuring 81 by 84 inches and the other measuring 24 by 60 inches. The areas of these rectangles are, respectively, 6,804 and 1,440 inches. The floor area of the room is therefore 44,712 (6,804 1,440) 44,712 8,244 36,468 square inches. You might want to divide by 144 to obtain the answer as 253.25 square feet. 12. Consider the walls as a set of rectangles, each of which is 7 feet (84 inches) high. Then starting at the upper left-hand corner and proceeding clockwise, the areas of the rectangles are 84 84 7,056 square inches 24 84 2,016 square inches 60 84 5,040 square inches 24 84 2,016 square inches 63 84 5,292 square inches 54 84 4,536 square inches Window 0 square inches 54 84 4,536 square inches 27 84 2,268 square inches Window 0 square inches 27 84 2,268 square inches 84 84 7,056 square inches 81 84 6,804 square inches 48 84 4,032 square inches Door 0 square inches 48 84 4,032 square inches The surface area of the walls is the sum of all these areas, or 56,952 square inches. You might wish to divide this figure by 144 to obtain the answer as 395.5 square feet. CHAP. 7 1. In an hour, a car going 35 mph will go 35 miles, and 36 minutes is 3/5 of an hour. Therefore, in 36 minutes, if the car travels at a constant speed, it will go 35 3/5 21 miles. 2. The answers are (a) Upstream speed is 10 2 8 mph. (b) Downstream speed is 10 2 12 mph.
530 Mastering Technical Mathematics 3. The answers are (a) To go 96 miles upstream takes 96/8 12 hours. (b) To go 96 miles downstream takes 96/12 8 hours. 4. The answers are (a) The upstream journey takes 12 hours. In this time, at a constant water speed of 10 mph, the riverboat burns 12 1/2 6 tons of fuel. (b) The downstream journey takes 8 hours. In this time, at a constant water speed of 10 mph, the riverboat burns 8 1/2 4 tons of fuel. 5. To make the downstream journey at a land speed of 8 mph, the water speed would have to be cut to 6 mph, or 60 percent (0.6) of its former speed. This would result in the boat burning 1/2 0.6 0.3 ton of fuel per hour. The journey of 96 miles at 8 mph would take 96/8 12 hours, which would burn 12 0.3 3.6 tons of fuel. That is 0.4 ton less than before. The speed reduction would save fuel. The fuel savings for the downstream journey would be 0.4/4 10 percent. Without the speed reduction, the round-trip fuel burned is 10 tons. With the downstream water speed reduction, 0.4 ton is saved going downstream but nothing is saved going upstream (assuming the boat operates in the same way as before during that part of the journey). The total fuel savings for the round trip is therefore 0.4/10 4 percent. 6. Suppose the water speed were cut by 2 mph going upstream, from 10 mph to 8 mph. Then the land speed would be reduced to 6 mph, and it would take the boat 96/6 16 hours to make the upstream journey. Assuming the fuel consumption decreased in proportion to the water speed reduction, there would be 0.5 0.8 0.4 ton burned per hour. In 16 hours the ship would burn 16 0.4 6.4 tons of fuel. Therefore, if the water speed upstream were reduced by 2 mph, the total amount of fuel burned would increase from 6 tons to 6.4 tons! This is a change of 100 0.4/6 6.667 (or 6-2/3) percent. 7. The value increases by $10,000, and the dividend paid on the initial investment is $50,000 0.05 $2,500. Therefore, the total amount made in cash is $10,000 $2,500 $12,500. This is $12,500/$50,000 0.25, or 25 percent. 8. The percentages of growth break down like this: First week (24 16)/16 8/16 0.5 50 percent Second week (36 24)/24 12/24 0.5 50 percent Third week (54 36)/36 18/36 0.5 50 percent Fourth week (81 54)/54 27/54 0.5 50 percent Entire four-week period (81 16)/16 65/16 4.0625 406.25 percent 9. See Fig. A-1. In the middle of the second week, the height is 30 inches as interpolated from the graph. 10. At 90 mph, a car travels 3 miles in 3/90, or 1/30, of an hour, which is 2 minutes. For the whole lap, this car takes 6 2 8 minutes. That translates to an average speed of 8 60/8 60 mph. At 120 mph, a car travels 3 miles in 3/120, or 1/40, of an hour; that’s 1.5 minutes. For the whole lap, this car takes 6 1.5 7.5 minutes. That translates to an average speed of 8 60/7.5 64 mph. 11. At 32 miles per gallon, the car will travel 18 32 576 miles on 18 gallons.
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Mastering Technical Mathematics
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Mastering Technical Mathematics Thi
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To Tim, Samuel, and Tony
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For more information about this tit
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Contents ix Real Vs. Imaginary Numb
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Contents xi Navigator’s Polar Vs.
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Contents xiii Law of Large Numbers
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Introduction This book is intended
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Copyright © 2008, 1999 by The McGr
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1From Counting to Addition We’ve
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From Counting to Addition 5 Figure
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From Counting to Addition 13 WEIGHT
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13. In the United States, common li
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2Subtraction Just as addition is co
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Subtraction 21 the hundreds column,
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Subtraction 23 MAKING CHANGE This i
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Subtraction 25 Figure 2-6 Two metho
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3Multiplication Suppose you go into
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Multiplication 29 In Fig. 3-2A, exa
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Multiplication 31 Figure 3-3 An exa
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Multiplication 33 Figure 3-6 It doe
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Multiplication 35 Figure 3-9 Subtra
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Multiplication 37 Figure 3-12 Multi
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Multiplication 39 3. Multiply the f
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4Division Before the days of electr
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Division 43 Figure 4-2 A pictorial
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Division 45 MULTIPLICATION CHECKS D
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Division 47 Figure 4-6 At A, we can
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Division 49 In the example of Fig.
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Division 51 7/10 0.7 17/20 0.85 3
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Division 53 QUESTIONS AND PROBLEMS
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5Fractions Chapter 4 presented some
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Fractions 57 This line of reasoning
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Fractions 59 493/23 21.xxx . . . (
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Fractions 61 The numbers in the ori
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Fractions 63 5/12 25/60 3/10 18/6
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Fractions 65 “limiting values”
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Fractions 67 QUESTIONS AND PROBLEMS
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6Plane Polygons So far, this book h
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Plane Polygons 71 WHAT IS A SQUARE?
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Plane Polygons 73 inches is not the
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Plane Polygons 75 Figure 6-7 Two wa
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Plane Polygons 77 AREA OF AN OBTUSE
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Plane Polygons 79 Figure 6-11 Some
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Plane Polygons 81 QUESTIONS AND PRO
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7Time, Percentages, and Graphs You
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Figure 7-2 A racecar driver can cal
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When a percentage expresses a chang
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Time, Percentages, and Graphs 89 Fi
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Time, Percentages, and Graphs 91 Fi
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5-mile track. Each graph gives you
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Part 2 ALGEBRA, GEOMETRY, AND TRIGO
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8First Notions in Algebra As proble
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First Notions in Algebra 99 Figure
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First Notions in Algebra 103 (a b)
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To recap, here are the rules if you
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4. Suppose x 5 and y 7. Then what
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9Some “School” Algebra In this
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Some “School” Algebra 115 DIMEN
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Some “School” Algebra 117 4 u
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Some “School” Algebra 119 20x
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Some “School” Algebra 121 Figur
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Figure 9-9 Sometimes it’s easier
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9. Adding 1 to the numerator and de
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10Quadratic Equations Simultaneous
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Quadratic Equations 129 Figure 10-3
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What about the “negative solution
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11 Some Useful Shortcuts Have you n
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Some Useful Shortcuts 143 FINDING S
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used to derive square roots to seve
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Some Useful Shortcuts 147 Figure 11
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Some Useful Shortcuts 149 Figure 11
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Some Useful Shortcuts 151 which sim
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12. In each of the following pairs
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12 Mechanical Mathematics In this c
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Mechanical Mathematics 157 Figure 1
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is at (a times t) feet per second.
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If the mass is 1 kg, the force is 9
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13 Ratio and Proportion Thus far, t
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Ratio and Proportion 175 PROPORTION
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In any triangle, its three interior
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Ratio and Proportion 179 Figure 13-
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14 Trigonometric and Geometric Calc
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Trigonometric and Geometric Calcula
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PROPERTIES OF THE ISOSCELES TRIANGL
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Suppose you name the equal pairs of
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Copyright © 2008, 1999 by The McGr
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15 Systems of Counting The decimal
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Systems of Counting 205 Figure 15-2
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Systems of Counting 207 OCTAL NUMBE
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Systems of Counting 209 DECIMAL BIN
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are shortcut methods for performing
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1. Find the decimal equivalent of t
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16 Progressions, Permutations, and
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Progressions, Permutations, and Com
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17 Introduction to Derivatives The
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Figure 17-2 The derivative of the f
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is negative when x is negative, bec
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This rule holds true for all expres
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Introduction to Derivatives 243 CIR
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Introduction to Derivatives 245 Fig
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18 More about Differentiation The p
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More about Differentiation 253 Figu
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More about Differentiation 255 by t
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More about Differentiation 257 MULT
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In the powers table, 0 n is always
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DERIVATIVE OF FUNCTION MULTIPLIED B
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More about Differentiation 263 QUES
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19 Introduction to Integrals Now th
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Introduction to Integrals 267 Figur
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(it subtracts from itself), so you
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Introduction to Integrals 275 VOLUM
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1. Find the derivatives of the prod
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20Combining Calculus with Other Too
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plotting nine points, but to get a
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In a situation like this, you have
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Combining Calculus with Other Tools
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These principal axes can be regarde
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e reaches 1, the ellipse “breaks
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etween the plane and the double con
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21Alternative Coordinate Systems Th
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Alternative Coordinate Systems 303
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perpendicular to the other two. The
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22Complex Numbers Long ago, humanki
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Multiplying repeatedly by i works t
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Complex Numbers 327 Figure 22-4 The
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Complex Numbers 329 When you cube i
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Complex Numbers 331 Figure 22-9 The
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Complex Numbers 333 Figure 22-11 At
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Complex Numbers 335 QUESTIONS AND P
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Part 4 TOOLS OF APPLIED MATHEMATICS
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23Trigonometry in the Real World Tr
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The absolute accuracy (in fixed uni
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Trigonometry in the Real World 343
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objects, even when they are observe
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Now find the distance PV by applyin
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7. What is the range of the aircraf
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24Logarithms and Exponentials A log
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Logarithms and Exponentials 361 Fig
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Now try an example in which neither
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Let’s put a number into this form
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Logarithms and Exponentials 369 10
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Suppose x is some real number. The
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To demonstrate this, let b e, x 2
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“Inv” key followed by a “ln
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and so on. A decrease of 1 order of
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25Scientific Notation Tutorial In s
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Another alternative multiplication
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line that covers 3 orders of magnit
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Scientific Notation Tutorial 387 (3
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Scientific Notation Tutorial 389 (5
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equality symbols. It is universally
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Sometimes, the outcome of determini
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26Vectors You’ve already seen a f
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Vectors 397 MULTIPLICATION OF A VEC
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Vectors 399 The values of the angle
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Vectors 401 If a is multiplied by a
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Vectors 403 Figure 26-5 Direction a
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Vectors 405 This formula works in a
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Vectors 407 You are told that these
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Vectors 409 4. Consider the two vec
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27Logic and Truth Tables To gain a
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are true. If any one of the compone
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antecedent is (P & C). The “impli
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Logic and Truth Tables 417 X Y X &
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Logic and Truth Tables 419 (X 1 Y)
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Logic and Truth Tables 421 The same
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classes—a certain way that multip
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Table 27-9B Derivation of truth val
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Logic and Truth Tables 427 A X Y X
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Logic and Truth Tables 429 X Y Z X
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Logic and Truth Tables 431 X Y Z X
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28Beyond Three Dimensions As you ha
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Beyond Three Dimensions 435 Figure
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a little later! It “lives” for
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Beyond Three Dimensions 441 IMPOSSI
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Beyond Three Dimensions 443 Displac
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Beyond Three Dimensions 449 NO PARA
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such as the pyramid, cube, and sphe
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2. If there are no pairs of paralle
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29A Statistics Primer Statistics is
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A Statistics Primer 457 Figure 29-2
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A Statistics Primer 459 PARAMETER A
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A Statistics Primer 463 of dice, th
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To determine µ sc , check the tabl
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A Statistics Primer 471 making it m
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30Probability Basics Probability is
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Probability Basics 475 Event 1 Even
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Probability Basics 477 As with math
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Page 506 and 507: Probability Basics 489 Because the
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Page 512 and 513: Final Exam Do not refer to the text
Page 514 and 515: Final Exam 497 (a) An even number t
Page 516 and 517: Final Exam 499 X Y Z ? F F F F F F
Page 518 and 519: Final Exam 501 (c) A harmonic progr
Page 520 and 521: Final Exam 503 (c) 1.8 10 6 km (d)
Page 522 and 523: Final Exam 505 47. Suppose an item
Page 524 and 525: Final Exam 507 Figure E-6 Illustrat
Page 526 and 527: Final Exam 509 (a) p q (b) p q 1
Page 528 and 529: Final Exam 511 (d) Roots of w (e) P
Page 530 and 531: Final Exam 513 82. Which of the fol
Page 532 and 533: Final Exam 515 (b) It decreases in
Page 534 and 535: Final Exam 517 95. In the situation
Page 536 and 537: Appendix 1 Solutions to End-of-Chap
Page 538 and 539: Solutions to End-of-Chapter Questio
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Solutions to End-of-Chapter Questio
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Figure A-30 Illustration for soluti
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Appendix 2 Answers to Final Exam Qu
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Appendix 3 Table of Derivatives Let
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Appendix 4 Table of Indefinite Inte
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Table of Indefinite Integrals 609 f
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Suggested Additional Reading Bluman
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Index absolute accuracy, 341 error,
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cosines, law of, 347-348 counter, m
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geometry elliptic, 449 Euclidean, 4
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exponential, 369-371 frequency, 255
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esonance, in system, 168-170, 255 r
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direction angles, 402-403 direction
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