Mathematics 10C

Year-end Review

McGraw-Hill Ryerson Version

Chapter 1 – Measurement Systems

1.1 SI Measurement

1. Determine the measurement of the bolt in millimetres and centimetres.

2. Determine the measurement of the block of wood in both millimetres and centimetres.

3. The Eiffel Tower was built in Paris in 1889. It has a height of about 300 m. An architectural technician makes a scale drawing of the tower using a scale factor of 0.0001.

a) How tall is the building, in centimetres?b) If, in her drawing, the tower were 6 cm high, what would the scale factor be?

4. If a trundle wheel covers 80 cm in one revolution of the wheel, what is the wheel’s diameter? Round your answer to the nearest tenth of a centimetre.

5. Convert the following measurements within the SI units:a) 42.6 cm to mb) 0.00152 km to cmc) 1500 mm to m

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1.2 Imperial Measurement

6. Determine the measurements in feet and inches of each of the readings illustrated on the tape measure.

7. Alexander Ovechkin of the Washington Capitals is listed as being 6 feet 2 inches tall? How many inches is this?

8. The elevation of Los Angeles, California is approximately 330 feet above sea level. How many yards is this?

9. Suzanne is redecorating a rectangular bedroom that is 12 feet 8 inches by 10 feet 4 inches. She wants to run a wallpaper border around the perimeter of the room, just below the ceiling.

a) What is the perimeter of the room?b) The wallpaper border comes in rolls that are 8 feet long. How many rolls of

wallpaper will Suzanne need (ignore any waste due to matching the pattern on the paper)?

10. The tree in the United States with the largest circumference is a Giant sequoia in Sequoia, National Park, California. It has a circumference of 85.3 feet. What is its radius, to the nearest tenth of a foot?

1.3 Converting Between SI and Imperial Systems

11. In basketball, the three-point field goal line is 23 feet 9 inches from the centre of the basket. How far is this in metres? How far is this in centimetres?

12. The distance to travel between Nashville, Tennessee and St. Louis, Missouri by car is approximately 515 kilometers. How many miles is this, to the nearest whole number?

13. The elevation of Edmonton, Alberta, is approximately 671 m above sea level. How many feet is this, to one decimal place?

14. Brian Kelly holds the Edmonton Eskimos team record for yards receiving in a career at 11 169 yards. How many metres is this, to the nearest whole number?

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15. Yakut cattle are a hardy breed of bovine that lives in northern Russia. They have an unusually large intestinal tract that, if uncoiled would reach approximately 52.5 m in length. To the nearest inch, how many inches is this?

16. The capybara of northern South America is the biggest rodent in the world. It can have a head-and-body length of up to 1.30 m.

a) How many inches is this, to the nearest inch?b) Express your answer in a) in feet and inches.

17. In the past, a common track and field race was the one-mile run. Now competitors run the 1500-m race. Which one is further and by how much?

18. Speed skiing is the sport of skiing downhill in a straight line as quickly as possible. It is one of the fastest non-motorized sports on land. The current world record is 251.4 km/h, held by Simone Origone, of Italy. How fast is this speed in miles per hour, to one decimal place?

19. The bridge in Cambridge, Massachusetts that leads from the MIT campus to Boston Proper is measured in smoots (so named after an unfortunate fraternity pledge named Oliver Smoot, who was used to measure the bridge in 1958). One smoot is equivalent to

feet. If the height of the bridge is 364.4 smoots, what is the measure of the bridge to the nearest tenth of a metre?

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Chapter 2 – Surface Area and Volume

2.1 Units of Area and Volume

20. Biff’s Carpet and Toupee Emporium is offering carpet at $2.58 per square foot. A private shop is selling the same carpet at $22.88 per square yard. Which store has the better price?

21. The largest flag in the world is a rectangular Israeli flag that was unveiled in 2007. The flag measures 660 metres by 100 metres.

a) What is the area of the flag in square metres?b) What is the area of the flag in square feet, to the nearest hundred square feet?

22. The West Edmonton Mall in Edmonton is one of the largest shopping malls in the world. It covers 5.2 million square feet. How many square metres does the mall cover?

23. The dimensions of an iphone 4 box are 8.5 cm by 14.5 cm by 7 cm. Determine the volume of the box in cubic inches, to one decimal place.

24. Within the banks of Stonehenge are a series of 56 pits known as the Aubrey Holes. These holes make a nearly perfect circle with a diameter of 94 yards 2 feet 6 inches. What is the area of this circle, in square metres?

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2.2 Surface Area

25. Find the surface area of each of the following:a)

b)

c)

d)

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26. The sides of a square-based pyramid are 18 cm. The slant height of the pyramid is 12 cm. Find the surface area of the entire pyramid (including the base) to the nearest whole number.

27. The Louvre is a famous art gallery in Paris, France. In 1989, a large glass square-based pyramid was built in front of the Louvre to cover the main entrance. Each wall of the pyramid is a triangle with a base of 35.4 m and a slant height of 27.9 m. The base of the pyramid is not made of glass. What is the surface area of the glass in the pyramid, to one decimal place?

28. The Canadarm used on the space shuttle is a cylinder that is 15.2 m long with a diameter of 0.38 m. What is its surface area, to one decimal place?

29. Which has the greater surface area and by how much: a cone with radius 7.1 cm and a slant height of 10.0 cm; or a square-based prism with sides 12.0 cm and a height of 8.2 cm?

30. Given the following information, find the surface area of the sphere, to the nearest tenth:a) radius = 3 cmb) diameter = 2.5 m

31. If a sphere has a surface area of , what is its radius, to the nearest tenth of a metre?

32. The Cheery Chocolate and Candy Company produces chocolate coated cherry spheres with a thin coating of sugar. The diameter of each sphere is 6.3 mm. What is the surface area to be coated?

33. The surface area of a softball is about 118 cm2 greater than the surface area of a baseball. The radius of a baseball is 3.7 cm. What is the radius of a softball, to the nearest tenth of a centimetre?

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2.3 Volume

34. Find the volume of each of the following:a)

b)

c)

35. Sometimes, grain is stored in a cone-shaped pile. The diameter of the cone is 29.2 m, while the vertical height of the cone is 12.7 m. What is the total volume of the cone, to the nearest tenth?

36. The Muttart Conservatory in Edmonton consists of two small pyramids and one large one. Each small pyramid has a base of length 19.5m (assume that it is a square base) and a vertical height of 18.0m. What is the volume of each small pyramid?

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37. A hobby club runs remote-controlled boats in a tank that is shaped as shown:

The radius of the tank is 2.2 m, while the length of the tank is 14.2 m. Solve for the total volume of the water inside the tank, to one decimal place.

38. Given the following information find the volume of the sphere to the nearest tenth:a) radius = 6.5 cmb) diameter = 5.2 cm

39. North America's largest butane storage tank is at Point Tupper, Nova Scotia. The tank is a sphere with an inside radius of 12.8 m. What volume of butane can the tank hold, to the nearest cubic metre?

40. Find the volume of a basketball which is inflated to a diameter of 24.2 cm, to one decimal place.

Chapter 3 – Right Triangle Trigonometry

3.1 The Tangent Ratio

41. Solve for the following, to four decimal places:a)b)c)d)e)f)

42. Find <A to the nearest degree:a) tan A = 0.7002b) tan A = 0.9657

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c) tan A = 2.1445d) tan A = 3.4874e) tan A = 0.51095

43. In each triangle, solve for the unknown to one decimal place (drawings are not to scale):

44. The foot of a ladder is on level ground 2.4 m from a wall. The angle formed by the ladder and the ground is 68. Calculate how high up the wall the ladder reaches.

45. Rectangle ABCD measures 6 cm by 3 cm. Find the measures of the two acute angles formed at A by the diagonal AC.

3.2 The Sine and Cosine Ratios

46. Solve for the following to 4 decimal places:a)b)c)d)e)

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f)

47. Find <A to the nearest degree:a) sin A = 0.61566b) sin A = 0.70711c) sin A = 0.22495d) cos A = 0.76604e) cos A = 0.87156

48. In each triangle, solve for the unknown to one decimal place (drawings are not to scale):

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49. In ABC, <B = 90 and AC = 25.0m. Find AB and BC if:a) <A = 31 b) <C = 78

50. A wheelchair ramp is 8.2 m long and rises 0.75 m. What is the angle of elevation of the ramp, to the nearest degree?

51. A kite has a string 150 m long. If the string makes an angle of 41 with the ground, find the height of the kite.

52. A 1.5 m hoe rests against the side of a garden shed. The angle the handle of the hoe forms with the ground is 71. How far up the wall of the shed does the hoe reach?

3.3 Solving Right Triangles

53. A sighting is made of a sailboat from a lighthouse. If the lighthouse is 123.5 m above the level of the water, how far, to the nearest tenth of a metre, is the sailboat from the shore if the angle of depression is 18?

54. The light on the Prince Shoal lighthouse is 25 m above the water level. From a position beside the light, the angle of depression of a sailboat is 12. How far is the sailboat from the lighthouse, to the nearest metre?

55. Find YZ, to the nearest tenth of a metre:

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56. Find EF, to the nearest tenth of a metre:

57. Find UV, to the nearest tenth of a metre:

58. From the window of one building, Sam finds the angle of elevation of the top of a second building is 41 and the angle of depression of the bottom of the building is 54. The buildings are 56 m apart. Find the height of the second building.

Chapter 4 – Exponents and Radicals

4.1 Square Roots and Cube Roots

59. What is the value of each expression?

a)

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b)

c)

d)60. Find each square root to two decimal places.

a)

b)

c)

d)

61. The square root of a number is 7. What is the number?

62. The area of a square clock is 19 m2. What are the measurements of its sides?

63. A square lot has an area of 79 m2. a) Calculate the length of each side to one decimal place.b) How much fencing, to one decimal place, would be required to enclose the

lot?

64. The diameter of a circle is given by the formula , where d is the diameter of the circle and A is the area of the circle. Find the diameter of a circle with an area of 155 cm2, to the nearest tenth.

65. Evaluate each of the following.

a)

b)

c)

d)

66. In one day a gold mine produces approximately 1600 cm3 of gold. If five day’s gold production is cast into a cube, how long is one of its sides?

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4.2 Integral Exponents

67. Simplify each expression.

68. Simplify each expression.

69. Evaluate each expression.

70. Evaluate. Leave all of your answers as powers using positive exponents.

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71. Simplify each of the following as far as possible. Leave your answer with positive exponents.

72. In house construction, the safe load, m, in kilograms, that can be supported by a

horizontal joist of length l metres is given by the formula: where t is the thickness and h is the height in centimetres. Find the safe load for a beam 4 m long when the beam thickness is 5 cm and its height is 10 cm.

73. The stopping distance of a vehicle on an icy road depends on its speed and its mass. A

typical formula for estimating the stopping distance, d, in metres is: , where s is its speed in kilometers per hour and m is its mass in tonnes. Estimate the stopping distance for a 2000 kg car travelling at 100 km/h. Note: 1 tonne = 1000 kg.

74. The pressure, P, in kilopascals, exerted on the floor by the heel of a shoe is given by the

formula: , where m is the wearer’s mass, in kilograms, and x is the width of the heel, in centimetres. Find the pressure exerted by a 60 kg woman wearing shoes with heels 2 cm wide.

4.3 Rational Exponents

75. Use the laws of exponents to simplify. Leave your answers with positive exponents.

76. The function can be used to determine the frequency, in cycles per second of a musical note that is x half-steps above a note with a frequency k. The frequency of a

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middle C on a piano is 262 cycles per second. Find the frequency of the C that is one octave (12 half-steps) higher.

77. The astronomer Johann Kepler found a formula which can be used to determine the number of Earth-days it takes each planet to travel once around the sun. The formula is:

, where R is the mean distance from the planet to the sun in millions of kilometers and N is the number of Earth-days. Determine the number of Earth-days in the year of Saturn if Saturn is 1420 million kilometers away from the sun.

4.4 Irrational Numbers

78. Express each power as an equivalent radical.

79. Express each radical as a power.

80. Express each mixed radical as an equivalent entire radical.

81. Express each entire radical as an equivalent mixed radical.

82. Arrange the following from least to greatest:

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83. The Japanese board game of Tai Shogi is an expanded version of chess. Like chess, it is played on a square board covered with small squares. If each small square has a side

length of 2 cm, the diagonal of the whole board measures cm. How many squares are on the board?

Chapter 5 – Polynomials

5.1 Multiplying Polynomials

84. Expand and collect like terms.

85. Use the distributive property to determine each product.

86. Multiply and then collect like terms.

87. A side of a cube is cm long. Write a polynomial expression for the volume of the cube and then expand and collect like terms.

88. Write a simplified expression for the area of the shaded region:

6n

n

8n

11n

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5.2 Common Factors

89. Identify the GCF of the following sets of terms.

a)

b)

c)

90. Factor the following polynomials.

a)

b)

c)

d)

e)

91. Factor the following polynomials.

a)

b)

c)

d)

5.3 Factoring Trinomials

92. Factor, if possible.

a)

b)

c)

d)

e)

f)

g)

h)

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93. Factor each of the following, if possible.

a)

b)

c)

d)

e)

f)

g)

h)

94. Determine two values of b that allow each expression to be factored.

a)

b)

95. The CN Tower in Toronto has an area that can be expressed as square

units. Factor to find binomials that represent the length and the width of the tower.

96. The area of a children’s playground measures square units.a) Factor the binomials that represent the length and the width of the playground.b) If a represents 13 m, what are the length and the width of the playground, in

metres.

5.4 Factoring Special Trinomials

97. Factor each binomial, if possible.

a)

b)

c)

d)

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98. Factor each trinomial, if possible.

a)

b)

c)

d)

99. Evaluate the following by using the difference of squares method. a) b)c)

100. If a and b are integers, find values of a and b such that is 21.

101. The area of a square can be given by the expression , where x represents a positive integer. Write a possible expression for the perimeter of the square.

Chapter 6 – Linear Relations and Functions

6.1 Graphs of Relations

102. The graph shows the speed of a dirt bike versus the distance traveled from the start line during one lap of a race. Explain one possible scenario, given the graph:

Speed

Time

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103. Each graph shows distance from school versus time. Describe a situation that could be represented by each graph:

a) b)

Distance Distance

Time Time

104. The graph below shows the distance from port versus time of three boats taking people to watch whales. The boats stop when the whales are sighted. Describe what each boat did:

Distance

from

port

time

105. Describe a practical situation that could be represented by each graph:

a) b)

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6.2 Linear Relations

106. The following table represents a series of points that can be plotted on the Cartesian Plane. Fill in the spaces on the table and plot the points on a graph

x y ordered pair

1 22 43 6417

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6.3 Domain and Range

107. Give the domain and range of each graph. Use words, a number line, interval notation, and set notation.

a)

b)

c)

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d)

6.4 Functions

108. Determine whether or not each relation is a function.a) (2,5), (4,3), (6,1), (8,-1), (9,-2)b) (3,2), (5,6), (6,8), (3,-2), (6,-4)c) (8,1), (7,1), (-3,1), (-4,1)

109. Determine whether or not each relation is a function.a) b)

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c) d)

110. If , find:

a) b) c) d)

111. If , find:

a) b) c) d)

112. Consider the function

a) What is the value of

b) Determine x so that

113. A typical adult dosage for an antihistamine is 24 mg. Young’s rule for

determining the dosage size for a typical child of age a is

. What should the dosage be for a typical 8-year-old child?

114. The function H described by can be used to predict the height, in centimetres, of a woman whose humerus is x cm long.

Predict the height of a woman whose humerus is 34 cm long.

115. The function A described by gives the area of an equilateral triangle with side s. Find the area when a side measures 6 inches.

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6.5 Slope

116. Determine the slope of the line which passes through the following points:a) (6, 8) and (10, 5)b) (0, 0) and (3, 5)c) (3, 5) and (-2, -1)d) (-2, -4) and (-3, -5)e) (3.7, 5.1) and (-1.5, 1.2)

117. The slope of a plane is 0.02. Assume the slope is constant. What is the vertical height of the plane after it has traveled a horizontal distance of 450 m?

118. Determine the slope of each line.a) b)

Chapter 7 – Linear Equations and Graphs

7.1 Slope-Intercept Form

119. Find the slope and the y-intercept of each line.a)

b) c)

120. Express each of the following in slope and y-intercept form.a) b) c) d)

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e)

121. Write the equation of each line in the form a) slope = 2;  y-intercept = 3b) slope = 3;  y-intercept =

c) slope = ;  y-intercept = 1

d) slope = ;  y-intercept =

122. Consider the equation . What is each value of b if a graph of the line passes through each point?

a) (4, 11)b) (-2, -9)

7.2 General Form

123. Express each of the following in general form, . a)

b)

c)

124. Determine the x- and y-intercepts of each line.a) b) c)

125. Write an equation of the line that passes through the given point and has the given slope. Express the equation in general form.

126. Write an equation of the line that passes through the given points. Express the equation in general form.

127. What is the value of the unknown parameter in each equation?

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a) passing through

b) passing through

c) passing through 7.3 Slope-Point Form

128. Write an equation of the line that passes through the given point and has the given slope. Express the equation in point-slope form.

129. Write an equation in point-slope form of the line through and . 7.4 Parallel and Perpendicular Lines

130. Given the slopes of the two lines, determine whether the lines are parallel, perpendicular, or neither.

a) m1 = 2; m2 =

b) m1 = ; m2 = c) m1 = 5; m2 = -5

d) m1 = ; m2 =

131. Find the slope of a line perpendicular to a line with the given slope:

a) b) c) undefined

132. The slopes of two parallel lines are and . Find the value of m.

133. The slopes of two perpendicular lines are and . Find the value of k.

134. Write an equation of the line through the point that is parallel to the line with the equation .

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135. Determine if the lines and are parallel, perpendicular, or neither.

136. Determine an equation for the line passing through and perpendicular to .

Chapter 8 – Solving Systems of Linear Equations Graphically

8.1 Systems of Linear Equations and Graphs

137. Solve each system of equations graphically and verify your solutions.

a) b)

c) d)

138. For which of these solutions is a solution?

a) b) c)

139. The total number of states in Germany and Austria is 25. Germany has 7 more states than Austria. Solve the following system of equations graphically to

find the number of states in each country.

140. Sea otters are found along the shores of the North Pacific. Sea otters were almost extinct in 1910, because they had been over-hunted for their fine, silky

brown fur. They are now protected by an international treaty. The two main populations of sea otters are along the coasts of California and Northern BC-Alaska. If n represents the approximate number of sea otters in the north, and s represents the approximate number of sea otters in the south, the following two equations show how these numbers are related:

Solve the system of equations to find the approximate number of sea otters in each population.

8.2 Modelling and Solving Linear Systems

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141. For the following questions, assign a variable to each unknown quantity. Then translate the problem into a system of linear equations that could be used

to determine each unknown. Do not solve the system.a) Three footballs and one soccer ball costs $155. Two footballs and

three soccer balls costs $220. Determine the cost of one football and the cost of one soccer ball.

b) Two shirts and one sweater costs $60. Three shirts and two sweaters costs $104. Determine the prices of one shirt and

one sweater.

c) At a sale, all CDs are one price and all tapes are another. Three CDs and two tapes cost $72. One CD and three tapes cost

$52. What are the prices of one CD and one tape?d) An aerobics club has an initial fee and a monthly fee. The cost for

a six-month membership of $220 and the cost for a one-year membership is $340. Write equations to determine the initial fee and the monthly fee.

e) Evan invested $2050 in the stock market. Part of the money was invested in a stock worth $5 per share and the rest was

invested in a stock worth $8 per share. He purchased 350 shares altogether. Write equations to determine the number of shares purchased at each price.

8.3 Number of Solutions for Systems of Linear Equations

142. Determine whether each linear system has infinitely many solutions, no solution, or exactly one solution. Justify your answers:

a)

b)

c)

143. One equation in a linear system is . The system has no solution. Determine whether or not the following equations could define the second

equation in the system.a) b) c) d)

Chapter 9 – Solving Systems of Linear Equations Algebraically

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9.1 Solving Systems of Linear Equations by Substitution

144. Solve each of the following by using the substitution method:

a)

b)

c)

d)

e)

145. The highest point in British Columbia, f metres above sea level is on Fairweather Mountain. The highest point in Manitoba, b metres above

sea level is on Baldy Mountain. The heights are related by the following system of equations:

Solve the system of equations by substitution to find the height of each mountain.

146. The sum of two numbers is 752 and their difference is 174. Use substitution to determine the two numbers

147. A chocolate manufacturer has found by consumer research the most popular mix of hard and soft-centered chocolates. The best profit is given by the

equations:

where s is the number of soft-centered chocolates and h is the number of hard-centered chocolates. Solve the system of equations by using substitution

to find the number of each kind of chocolate.

9.2 Solving Systems of Linear Equations by Elimination

148. Solve each of the following by using the elimination method:

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a)

b)

c)

d)

e)149. At the cafeteria, three hamburgers and three Cokes cost $9.00. Two hamburgers

and one Coke cost $4.75. Use the elimination method to determine the cost of one hamburger and the cost of one Coke.

150. The receipts from 550 people attending a play were $9184. The tickets cost $20 for adults and $12 for students. Use the elimination method to find

the number of adults and student tickets sold.

151. Solve the following system of equations by elimination and explain the result.

9.3 Solving Problems Using Systems of Linear Equations

152. An aircraft travels 5432 km from Montreal to Paris in 7 hours and returns in 8 hours. The wind speed is constant. Determine the wind speed and the

speed of the aircraft in still air.

153. A plane flew 9000 km from Seoul, South Korea to London, England with the wind in 10 hours. The return flight against the wind took 11.25 hours.

Find the wind speed and the speed of the plane in still air.

154. Sir John A. Macdonald and William Lyon Mackenzie King are two of the longest serving Prime Ministers of Canada. King’s three periods in office totalled

three more years than Macdonald’s two periods. Together, Macdonald and King served for 41 years. Determine the number of years that Mackenzie King served in office.

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