TUTO R IAL 1.3 Measurement Conversions For all of the following exercises, use technology or a conversion table. 1. Convert the following measurements using the indicated units. a) 5 feet 6 inches; inches b) 14 yards; feet c) 1500 cm; metres d) 135 km; metres e) 78 inches; feet f) 5280 yards; miles g) 2 square yards; square feet h) 750 mm 2 ; square centimetres 2. Convert the following measurements using the indicated units. a) 5 feet 10 inches; centimetres b) 27 cm; inches c) 5 imperial gallons; litres d) 150 miles; kilometres e) 3 yards; metres f) 4 square miles; square kilometres g) 25 cm 2 ; square inches h) 15 pounds; kilograms i) 4 kg; pounds 3. A football field is 110 yards long measured from goal post to goal post. A rugby field is 100 m long. Which field is longer 4. An apprentice jockey must have a mass less than 130 pounds. Brittney has a mass of 65 kg. Is she eligible for this job Justify your answer. 5. The doorway to Jai s office is 2 feet 8 inches wide. He wishes to purchase a desk 80 cm wide. Can the desk pass through his office doorway 6. Which is the better buy for T-bone steak: $6.98/lb or $14.98/kg 7. The living room of an apartment has dimensions 14 feet by 13 feet and the dining room has dimensions 13 feet by 9 feet. Determine: a) the total area of the two rooms b) the cost of carpeting both rooms at $26.50 per square yard c) the cost of carpeting both rooms at $29.99/m 2 d) Suppose both carpets are of the same quality. Which is the better buy and by how much 8. The speed limit for a highway in Canada is given in km/h. In the United States, the speed limit is given in mph (miles per hour). a) Complete the table below converting the speed limit from mph to an equivalent speed limit in km/h to the nearest whole number. Speed Limit (mph) 60 55 65 40 25 Speed Limit (km/h) ETRA PRACTICE EERCISES 1
b) Draw a scatterplot of the data. Plot speed limit in mph on the horizontal axis and speed limit in km/h on the vertical axis. Draw a line to represent this relationship. c) Use your graph to convert a speed of 50 mph into km/h. d) Use your graph to convert a speed of 110 km/h into mph. e) If the posted speed limit is 35 mph and you are travelling 55 km/h, could you get a speeding ticket Explain why or why not. Answers: 1. a) 66 inches b) 42 feet c) 15 m d) 135 000 m e) 6.5 feet f) 3 miles g) 18 square feet h) 7.5 cm 2 2. a) 177.8 cm b) 10.6 inches c) 22.7 L d) 241.4 km e) 2.7 m f) 10.4 km 2 g) 3.9 square inches h) 6.8 kg i) 8.8 pounds 3. Football field; 110 yards = 100.6 m 4. No; 65 kg = 143 pounds 5. Yes; the doorway is 2 8 = 32 = 81.28 cm wide; the desk is 80 cm wide. 6. $14.98/kg; $6.98/pound is approximately equal to $15.36/ kg. 7. a) 299 square feet b) $880.39 c) $833.06, d) The better buy is $29.99/m 2 by $47.33. 8. a) Speed Limit (mph) Speed Limit (km/h) b) 110 60 97 100 55 89 65 105 90 40 25 64 40 0 10 20 30 40 50 60 70 Speed Limit (mph) c) 80 km/h d) 68 mph e) No, 55 km/h is equivalent to 34.2 mph, which is below the speed limit. Speed Limit (km/h) 80 70 60 50 40 30 20 10 2 ETRA PRACTICE EERCISES TUTORIAL 1.4 Precision and Accuracy in Measurement 1. Explain whether it is reasonable to use each of the following measuring instruments for the given measurement. a) a ruler with a precision of 1 mm to measure the width of your palm b) a ruler with a precision of 1 mm to measure the length of the parking lot c) a vernier caliper that measures to the nearest 0.01 cm to measure the diameter of a golf ball d) a micrometer that measures to the nearest 0.01 mm to measure the thickness of a dime e) a scale with a precision of 1 kg to find the mass of an orange f) a scale with a precision of 1 kg for the mass of a small car
g) a measuring cup with a precision of 1 cup to measure the flour to 4 bake a cake h) a measuring cup with a precision of 1 cup to measure the salt to 4 bake a cake i) a watch with a second hand to measure how long a traffic light stays red j) a watch with a minute hand to measure the period (1 swing back and forth) of a 50-cm pendulum 2. A framer uses a ruler to determine the length and width of frame for a picture. a) Is it important for the ruler to be accurate Why or why not b) The framer decides to use a tape measure marked in centimetres to measure the frame and then determine where the cuts should be made. Explain why this could result in the frame not fitting the picture. 3. Use four s on each diagram below to show: a) good precision, good accuracy b) good precision, bad accuracy c) bad precision, good accuracy d) bad precision, bad accuracy a) b) c) d) 4. At the 1996 Olympic Games, Donovan Bailey won the 100-m sprint in a time of 9.84 s, setting a world record. a) Explain why a timer with precision to the nearest tenth of a second is not adequate for the 100-m sprint. b) Explain why a timer with an uncertainty of ± 0.05 s would not be adequate for timing the 100-m sprint. 5. A student wishes to find out whether there is significant variation in the masses of bags of candies that are each labelled 60 g. Using a balance scale precise to the nearest 0.1 g, she measures the mass of 1 bag of candy and then measures the mass of 10 bags of candy. Is this balance scale an appropriate measuring instrument for this experiment Explain. 6. When a carpet layer and his assistant determined the length of a room, a measurement error of 1 cm occurred. Would you consider this a significant error Why or why not 7. Which is most precise: a bathroom scale that measures to the nearest kilogram, a decigram balance scale that measures to the nearest 0.1 g, or a balance with a precision of 1 mg Explain. 8. Using a vernier caliper, a student determined the diameter of a glass to be 7.32 cm. Using a ruler, the student found the height of the glass to be 17.4 cm. ETRA PRACTICE EERCISES 3
a) What is the precision associated with each measuring device b) Suppose the true measurements are known to be 7.45 cm and 17.5 cm respectively. Explain which measurement is more accurate. 9. Identify three careers where precision and accuracy are critical to the performance of regular duties. Answers: 1. a) Yes, a palm is about 90 mm wide. b) No, a parking lot is too long. c) Yes, the diameter of a golf ball is approximately 4.3 cm. d) Yes, a dime has appropriate thickness. e) No, an orange is not heavy enough. f) Yes, a car is heavy enough. g) Yes, if the amount of flour required is more than 1 cup. h) No, a cake 4 takes very little salt. i) Yes. The average time a traffic light stays red is less than 2 min. j) No, the time required for a period is too short. 2. a) Answers may vary. If the framer is cutting her own frames then the accuracy of the ruler is not important, provided she uses the same ruler to measure the picture and to measure the frames she cuts. If the framer buys the frames elsewhere, then her ruler must be accurate or the frame might not fit the picture. b) Since the tape measure has a precision of 1 cm the uncertainty of each measurement is ± 0.5 cm. If the picture was really 12.3 cm long, the framer might cut 12.5 cm due to the precision of the measure. 3. a) b) c) d) 4. a) If the timer was precise only to the nearest tenth of a second, several runners could finish the race with the same time and no winner could be declared. b) A timer with an uncertainty of 0.05 s has a precision of one-tenth of a second. It is not adequate for the 100-m sprint, which is timed to the nearest one-hundredth of a second. 5. Since one bag of candy has a mass of 60 g this balance scale is appropriate. An uncertainty of ± 0.05 g is insignificant 6. An error of 1 cm is significant because there could be a gap of 1 cm between the carpet and the wall. 7. A balance with a precision of 1 mg is most precise since the divisions on its scale are smaller than those of the other two measuring devices.the smaller the units of measurement, the lower the uncertainty and the greater the precision 8. a) The precision for the diameter of the glass is 0.01 cm. The precision for the height of the glass is 0.1 cm. b) Since the measure of the height of the glass differs from the actual measure by a smaller amount, this measure is more accurate. 9. Answers may vary. Some suggestions are: carpenter, engineer, architect, nurse, pro basketball player, dentist, doctor. TUTORIAL 1.5 V olume of a Sphere Reminder: When you calculate the volume of a sphere, always use the value of π stored in your calcuator. 1. Find the volume of each sphere described below. a) a baseball with a radius of 3.7 cm b) a marble with a diameter of 15 mm c) a balloon with a diameter of 1.27 m d) a soccer ball with a radius of 10.9 cm 4 ETRA PRACTICE EERCISES
2. Calculate the volume of each ball. a) a tennis ball with radius 3.6 cm b) a squash ball with diameter 4.0 cm 3. A round bubble gum has a diameter of 26 mm. Calculate the volume of the gum to the nearest tenth of a cubic millimetre. 4. a) Calculate the volume of air in a spherical balloon with a diameter of 20 cm. b) If air was pumped into the balloon until the diameter was doubled, what is the new volume of the balloon c) How much more air was pumped into the balloon d) As the diameter of the balloon doubled, by how many times did the volume increase 5. The radius of the moon is approximately 1 that of Earth. The mean 4 radius of Earth is 6365 km. Calculate: a) the radius of the moon b) the volume of the moon 6. A rubber ball has a circumference of 13.2 mm. a) Use the formula for the circumference of a circle to determine the radius of the rubber ball. b) Determine the volume of the rubber ball. 7. A giant jawbreaker has a circumference of 9.4 cm. a) Use the formula for the circumference of a circle to determine the diameter of the jawbreaker. b) Use the diameter to calculate the volume of the jawbreaker. 8. At the Old Ice Cream Shop, you can purchase an ice cream cone with two small scoops of ice cream each with a radius of 3 cm. For the same price, you could buy a cone with one large scoop of ice cream with a radius of 4 cm. a) Calculate the volume of ice cream in each case. b) What is the better bargain c) Why might people choose the option that is not the better bargain 9. Lorraine makes and sells candles in various shapes and sizes. She has found that the most popular candles are a sphere with a radius of 5 cm, a cylinder with a radius of 4 cm and a height of 12 cm, and a rectangular candle with a length of 5 cm, width of 5 cm, and a height of 15 cm. a) Calculate the volume of wax needed to make each candle. b) Lorraine prefers to make the candle that requires the least amount of wax. Which candle is this Answers: 1. a) 212.2 cm 3 b) 1767.1 mm 3 c) 1.07 m 3 d) 5424.6 cm 3 2. a) 195.4 cm 3 b) 33.51 cm 3 3. 9202.8 mm 3 4. a) 4188.8 cm 3 b) 33 510.3 cm 3 c) 29 321.5 cm 3 d) 8 times 5. a) 1591.25 km b) 1.7 10 10 km 3 6. a) 2.1 mm ETRA PRACTICE EERCISES 5
b) 38.8 mm 3 7. a) 3.0 cm b) 14.1 cm 3 8. a) Two scoops = 226.2 cm 3 ; one scoop = 268.1 cm 3 b) One large scoop c) Might purchase two small scoops for variety of flavours. 9. a) Sphere = 523.6 cm 3 ; cylinder = 603.2 cm 3 ; rectangle = 375 cm 3 b) Rectangular candle TUTORIAL 1.6 Surface Area of a Sphere 1. Calculate the surface area of the spheres described below. Give each answer to the nearest tenth of a unit. a) a baseball with a radius of 3.7 cm b) a marble with a diameter of 15 mm c) a balloon with a diameter of 1.27 m d) a billiard ball with a radius of 30.75 mm 2. Calculate the surface area of each object. a) a tennis ball with radius 3.6 cm b) a squash ball with diameter 4.0 cm 3. a) Calculate the surface area of a spherical balloon with a diameter of 20 cm. b) Suppose air was pumped into the balloon until the diameter was doubled. What is the new surface area of the balloon c) By how much did the surface area increase d) As the diameter doubled, by how many times did the surface area increase 4. The radius of the moon is approximately 1 that of Earth. The mean 4 radius of Earth is 6365 km. Calculate the radius of the moon and then the approximate surface area of the moon. 5. The radius of a soccer ball is 10.9 cm. a) Calculate the surface area of the ball. b) Thirty-five percent of the surface of the ball is painted black and the remaining surface is painted white. Calculate the area of the ball that is painted black. c) What area is painted white 6. A beach ball is designed using equal segments of six different colours, one of which is pink. The ball is filled to a diameter of 60 cm. a) Calculate the surface area of the ball. b) How much of the surface is pink c) What percent of the surface is pink 7. A rubber ball has a circumference of 13.2 mm. a) Use the formula for the circumference of a circle to determine the radius of the rubber ball. b) Find the surface area of the rubber ball. 6 ETRA PRACTICE EERCISES
8. A giant jawbreaker has a circumference of 9.4 cm. a) Use the formula for the circumference of a circle to find the diameter of the jawbreaker. b) Use the diameter to calculate the surface area of the jawbreaker. 9. A town s water reservoir is in the shape of a sphere with a radius of 6 m. The town is planning to paint the reservoir using paint that costs $26.99 per can. One can of paint covers approximately 39 m 2. a) What is the surface area of the reservoir b) How many cans of paint are required if the reservoir needs only one coat c) How much will the paint cost 10. Lorraine makes candles in several shapes and sizes. She makes the candle out of a cheaper clear wax and then dips the finished product in a more expensive coloured wax. Out of the same amount of clear wax, Lorraine can make either a cube with a length of 14.5 cm or a sphere with a diameter of 18 cm. a) Calculate the surface area of each shape. b) What shape of candle would require the least amount of coloured wax Answers: 1. a) 172.0 cm 2 b) 706.9 mm 2 c) 5.1 m 2 d) 11 882.3 mm 2 2. a) 162.9 cm 2 b) 50.3 cm 2 3. a) 1256.6 cm 2 b) 5026.5 cm 2 c) 3769.9 cm 2 d) 4 times 4. radius 1591.25 km; 3.18 10 7 km 2 5. a) 1493.0 cm 2 b) 522.6 cm 2 c) 970.4 cm 2 6. a) 11 309.7 cm 2 b) 1885.0 cm 2 c) 16.6 % 7. a) 2.1 mm b) 55.5 mm 2 8. a) 3.0 cm b) 28.3 cm 2 9. a) 452.4 m 2 b) 12 cans c) $323.88 10. a) Cube = 1261.5 cm 2 ; sphere = 1017.9 cm 2 b) Sphere TUTORIAL 1.7 Square Roots and Cube Roots 1. Use a calculator to determine the approximate value of each of the following numbers. Round your answer to 2 decimal places. a) 10 b) 4.19 3 c) 24 3 d) 13.27 e) 3 0.001 f) 3 π 2. a) 49 = 7. Verify this answer by multiplying. 3 b) 27 = 3. Verify this answer by multiplying. c) Find 4 16 and verify your answer. 3. Solving an equation algebraically is a process of undoing operations. We undo an operation by doing the opposite operation. a) To undo multiplication, you use the operation. b) To undo addition, you use. c) To undo squaring, you use. d) To undo cubing, you use. e) To undo a cube root, you use. ETRA PRACTICE EERCISES 7
Note: For exercises 4 10, you will need the formulas for volume and surface area. It would be useful to prepare a reference table of these formulas. 4. The surface area of a peach is 154 cm 2. What is the radius 5. A cardboard box in the shape of a cube has a volume of 4 cubic feet. What is the length of one side of the box 6. A beach ball holds 5 L of air. Find the radius of the beach ball to the nearest tenth of a centimetre. (1 ml = 1 cm 3 ) 7. A spherical bath bead is entirely filled with 3 ml of bubble bath. Find the diameter of the bath bead to the nearest tenth of a millimetre. 8. Suppose the surface area of a globe is 460 square inches. Will the globe fit into a shelf space that has a width of 14 inches 9. The surface area of a ball is 255.5 cm 2. a) Find the diameter of the ball. b) Use your answer from part a to determine the circumference of the ball. 10. For a birthday party, Darci made candies by dipping cherries into chocolate. Each cherry has a volume of approximately 2 cm 3. a) Find the diameter of each cherry. b) Use your answer from part a to determine the surface area of chocolate covering one cherry. c) What is the surface area of chocolate covering 100 cherries 11. The size of a bowling ball is larger for 10-pin bowling than for 5-pin bowling. With 2700 cubic inches of a plastic compound, a manufacturer can either make 8 large bowling balls or 41 small bowling balls. a) Find the volume of each size of bowling ball. b) Find the diameter of each size of bowling ball. c) Find the surface area of each size of bowling ball. Answers: 1. a) 3.16 b) 2.05 c) 2.88 d) 2.37 e) 0.10 f) 1.46 2. a) 7 7 = 49 b) 3 3 3 = 27 c) 2; 2 2 2 2 = 16 3. a) Division b) Subtraction c) Square root d) Cube root e) Cubing 4. 3.5 cm 5. 1.59 feet 6. 10.6 cm 7. 17.9 mm 8. Yes, diameter of the globe is 12.1 inches. 9. a) 9.0 b) 28.3 cm 10. a) 1.56 cm b) 7.677 cm c) 767.7 cm 2 11. a) Volume of 10-pin ball = 337.5 cubic inches; volume of 5-pin ball = 65.9 cubic inches. b) Diameter of 10-pin ball = 8.6 inches; diameter of 5-pin ball = 5.0 inches. c) Surface area of 10-pin ball = 232.4 square inches; surface area of 5-pin ball = 78.5 square inches. 8 ETRA PRACTICE EERCISES
TUTORIAL 1.8 Enlarging and Reducing Figures 1. Melfort is 285 km north of Regina. On a map of Saskatchewan, the distance between Melfort and Regina is 9.5 cm. a) Determine the linear scale factor of the map. b) Yorkton is 2.2 cm from the Manitoba border on the map. Determine the actual distance from Yorkton to the border. c) Suppose the area of Saskatchewan on the map is about 724 cm 2. What is the approximate area of the province in km 2 2. The radius of a softball is 1.3 times the radius of a baseball. How many times larger is the volume of a softball than that of a baseball 3. The dimensions of a rectangular photograph were doubled in length. a) What is the ratio of the original area to the area of the enlargement b) Is this ratio the same as the scale factor Explain you answer. 4. The radius of a sphere is tripled. a) By what factor does the surface area increase b) By what factor does the volume increase 5. The area of a cattle pen is 140 m 2. To make space for new calves, the sides of the pen are doubled in length. a) By what factor does the area increase b) What is the area of the new pen 6. A living room window has an area of 1 m 2. The homeowner wants to replace the window with one of proportionate size that will have an area of 4 m 2. By what scale factor must each dimension of the window be multiplied to produce the new area 7. A large scoop of ice cream has a volume of 64 cm 3 and a small scoop of ice cream has a volume of 27 cm 3. What is the ratio of the radius of the large scoop to the radius of the small scoop 8. You can use a photocopier to decrease the area of a picture. a) Suppose the area was decreased to 64% of the original. By what factor were the dimensions changed b) Suppose the original picture took all of the space on an 8 1 2 14" sheet of paper. Would the reduced picture fit onto an 8 1 2 11" sheet of paper 9. For a birthday party, a balloon was filled with 2700 cubic inches of helium. By the next day, air had leaked out until there were only 800 cubic inches remaining. a) By what factor did the diameter of the balloon decrease b) By what factor did the surface area decrease ETRA PRACTICE EERCISES 9
10. At a movie theatre, a small box of popcorn costs $2.25. A larger box with dimensions 2 times that of the smaller box sells for $6.75. Which box of popcorn is the better buy Justify your answer. 11. A packaging company makes rectangular juice boxes. The regular size box has a capacity of 250 ml. A large box has a capacity of 500 ml. a) By what factor is the volume of the large box greater than that of the regular size box b) By what factor must you multiply the dimensions of the regular box to increase its capacity to that of the 500 ml box Answers: 1. a) 1 cm = 30 km b) 66 km c) 651 600 km 2 2. 2.197 times 3. a) 1:4 b) No the scale factor is 4 because it is the ratio of the new measurement to the old 1 measurement. 4. a) 9 b) 27 5. a) 4 b) 560 m 2 6. 2 7. 4:3 8. a) The new dimensios are 8 10 = 4 times the original, so the dimensions were changed by a scale 5 factor of 4 5. b) No. If you multiply the dimensions of 8.5" and 14" by 4 you find 5 that the new picture requires a length of 11.2". 9. a) 2 3 b) 4 10. If the dimensions 9 increase by a factor of 2, the volume increases by a factor of 2 3 or 8. In the larger box of popcorn then, you get 8 times as much popcorn for 3 times the price, making it the better buy. 11. a) 2 3 b) 2 or 1.26 times. 10 ETRA PRACTICE EERCISES