MATH on the Job
Old photographs and plane tickets become a personal artistic project when arranged on the pages of a scrapbook. Tamye Dunbar runs a stamping and scrapbooking company in Edmonton, Alberta. Traditional or electronic, scrapbooks can be made by pasting keepsakes onto book pages or generated using computer software. Scrapbooking is a million-dollar industry. As a hobby, it recently equalled golf in popularity.
tamye Dunbar supplies her clients with scrapbooking and stamping materials.
Scrapbooking supplies include feathers, stickers, ribbon, and coloured, textured paper. People often choose to base a scrapbook on a colour scheme, as well as an event, such as a birthday, wedding, or vacation.
Tamye’s customers order scrapbooking and stamping materials from her, which she ships in several different-sized boxes.
She determines which size of box to use based on how many materials she needs to ship.
As a business person, Tamye determined that her most popular
item is a greeting card kit. Her customers receive all the materials they need to make original greeting cards for special occasions, such as weddings or holidays. The greeting card kit comes in a box that is in the shape of a cube with a side length of 10 cm. If she needs to ship 125 of these items to one customer, which of the boxes below should she use?
50 cm
Every day, in many of your activities, you use volume and capacity without really thinking about them. Some examples of this are judging
volume: the measure of the space a three-dimensional object occupies
capacity: the amount a three-dimensional object can hold
• which pot will hold your can of soup
• how many boxes can fit in the back of a car
• how much gasoline you need to fill your gas tank
• how much clothing you can fit into a suitcase
139 Chapter 3 Surface Area, Volume, and Capacity
The volume of an object is the amount of space an object takes up (in three dimensions).
these containers have different capacities.
Because the world is three-dimensional, all objects have volume. Think of a sheet of paper—one sheet may not have much thickness, but when you stack 200 sheets, their volume becomes noticeable.
Volume is measured in cubed units, such as m3, in3 or ft3, reflecting the fact that volume is measured in three dimensions.
Capacity is closely related to volume. Capacity is the amount of material that can be contained in a hollow volume. A brick has volume in that it occupies space.
A box has volume in that it occupies space, but it also has capacity because it can contain another material
or other objects within it. Hollow objects have volume and capacity, but solid objects have only volume.
Capacity is often measured in units such as litres or gallons.
dIsCuss the Ideas
CalCulatIng the voluMe of pavIng stones
Julia is stacking small cubic paving stones on a pallet. One layer is 10 stones long and 10 stones wide, forming a square.
1. How many paving stones are in one layer?
2. Julia now stacks four more layers on top, so there are now a total of five layers. How many cubic paving stones are on the pallet?
3. If the edge length of each paving stone is one unit of length, what formula expresses the total number of paving stones?
4. What formula expresses the volume of the paving stones?
MathWorks 11 140
example 1
Luis sells different sizes of fish tanks in his pet store.
60 cm 20 cm
20 cm
Tank 1
40 cm 20 cm
20 cm
Tank 2
20 cm 20 cm
20 cm
Tank 3
a) How much water will be needed to completely fill each tank?
b) Look at tanks 2 and 3. How many dimensions have changed? By how much? How does the volume of tank 2 compare with that of tank 3?
c) Look at tanks 1 and 3. How many dimensions have changed? By how much? How does the volume of tank 1 compare with that of tank 3?
d) One litre equals 1000 cubic centimetres. Convert the volumes of the fish tanks to capacity in litres.
before purchasing an aquarium and fish from a pet store, ensure that you have enough space at home to house them safely.
solutIon
a) Tank 1: (60 × 20 × 20) = 24 000 cm3 Tank 2: (40 × 20 × 20) = 16 000 cm3 Tank 3: (20 × 20 × 20) = 8000 cm3
b) One dimension, the length, has changed. It has doubled. Tank 2 has twice the volume of tank 3.
c) One dimension, the length, has changed. It has tripled. The volume of tank 1 is three times the volume of tank 3.
d) Tank 1 24 000
1000 = 24 L Tank 2
16000 1000 = 16 L Tank 3
8000 1000 = 8 L
141 Chapter 3 Surface Area, Volume, and Capacity
example 2
Indoor climbing walls are built to mimic cliffs and steep slopes, and many rock climbers practise on them. Shapes, called climbing grips, are inserted into the wall for climbers to grasp. Some climbing grips are made of polyester resin that is poured into molds of different shapes. A mold for a cylindrical grip has a radius of 1.5 inches and a height of 3 inches. The machine that pours the polyester resin must be programmed by a technician to pour the amount that will fill the volume of the mold. What is the volume of the mold?
this rock climber has just finished scaling a 30-foot climbing wall. Climbing grips of different shapes are inserted into the wall for climbers to grip or step on as they ascend and descend.
solutIon
Use the formula for the volume of the mold:
area of base = πr 2 area of base = π(1.5)2 area of base = 2.25π
volume of mold = area of base × height volume of mold = 2.25π × 3
volume of mold = 6.75π volume of mold ≈ 21.2
The volume of the mold is 21.2 cubic inches.
ACtIVIty 3.5
voluMe of an oblIque prIsM
Tatyana and Sherri have been asked to find the volume of the oblique rectangular prism below.
8.5 cm
34 cm 5 cm
Sherri doesn’t think that it is possible to find the volume since it is not a rectangular prism. Tatyana remembers that you can find the area of a parallelogram by turning it into a rectangle.
8.5 cm
With a partner, decide if you can find the volume of an oblique rectangular prism using the same method. Be prepared to defend your position to the rest of your class.
MathWorks 11 142
dIsCuss the Ideas
InvestIgatIng voluMe
Julia is a landscaper. She will be planting flowers in 15 planters. There are three sizes of planters she can use. She has to determine how much soil will go into each so she knows how much soil to buy. The planters come as three cubic boxes, one with 1-ft sides, a second with 2-ft sides, and a third with 4-ft sides.
1. Determine the volume of each box.
1′
1′
1′
2′
2′
2′
4′
4′
4′
2. As the side length is doubled, what happens to the volume? Explain why you think this is the case.
3. If she has five of each size of planter, how much soil, in cubic feet, does she need?
the concrete planters outside this popular cafe in Victoria, bC, were fabricated by Mackay Precast, a company based in nanaimo, bC, that specializes in making precast concrete planters, picnic tables, benches, and garbage containers for clients across north America.
143 Chapter 3 Surface Area, Volume, and Capacity
example 3
A car’s engine is a complex machine consisting of many moving parts all working in unison.
Engine displacement is the volume swept by the pistons in an engine’s
cylinders. The following formula can be used to calculate engine displacement.
engine displacement = π
(
bore2)
2(stroke)(number of cylinders)The bore is the diameter of the engine’s cylinder. The stroke is the distance that the piston moves in the cylinder.
A 4-cylinder engine has a bore of 75.5 mm and a stroke of 82 mm.
a) What is the engine displacement in cm3?
b) Engine displacement is commonly stated in litres. What is the
displacement, in litres, of this engine? (Hint: One litre equals 1000 cm3.) c) Until recently, the engine displacement of cars manufactured in the United
States was stated in cubic inches. What is the displacement, in cubic inches, of the engine in part a)? (Hint: One inch equals 2.54 cm.) d) A car with a larger engine displacement generates more power than one
with a smaller engine displacement. How would the displacement change if the stroke were doubled?
e) How would the displacement change if the bore were doubled?
solutIons
a) engine displacement = π
(
7.552)
2(8.2)(4)engine displacement ≈ 1468
The engine displacement is 1468 cm3. b) engine displacement = 1468
1000 engine displacement = 1.468 The displacement is 1.468 L.
c) 1468
2.54 × 2.54 × 2.54 ≈ 89.6 The displacement is 89.6 in3.
d) Stroke appears as one dimension in the volume calculation; therefore, doubling the stroke would double the volume, or displacement.
e) Bore (or twice the radius) appears twice in the volume calculation (bore)2. Therefore, doubling the bore would increase the displacement by a factor of 4.
2 × 2 = 4
MathWorks 11 144
buIld your skIlls
Swimming helps to build muscle strength and endurance.
1. A community swimming pool is 100 feet long and 50 feet wide. It is 7 feet deep at the deep end and has a beach entry (0 feet deep) at the shallow end. The bottom of the pool has a constant slope.
a) Sketch the shape of the pool and label the dimensions.
b) What is the water capacity of the pool in cubic feet and in US gallons?
Remember that one cubic foot equals 7.48 US gallons.
c) What is the surface area of the inside of the pool?
Long cylinders of metal called rebar are used to reinforce building walls.
2. Concrete foundation walls for houses usually have a T-shaped bottom, called a footing. The footing spreads the weight of the house over a larger surface area to prevent settling (sinking) of parts of the house. The width of a footing is typically twice the width of the foundation wall. The footing thickness is equal to the width of the wall.
2 w
w w
a) The foundation wall of a house is 8 inches wide and the height of the wall and footing together is 4 feet. What volume of concrete, in cubic feet, is needed for a one-foot length of wall?
b) Concrete is usually ordered in cubic yards. One cubic yard equals 3 ft ×3 ft ×3 ft, or 27 ft3. How many cubic yards of concrete are needed for a 25-foot section of foundation wall?
145 Chapter 3 Surface Area, Volume, and Capacity
bakers must store all of their dry goods, such as flour, dried fruit, and spices, in airtight containers to prevent spoilage.
3. A bakery stores flour in a cylindrical bin 70 cm high and with a diameter of 50 cm.
a) What volume of flour does the bin hold?
b) The bakery orders flour in 20-kg sacks. Each sack is approximately 46 cm wide, 80 cm long, and 15 cm thick.
How many sacks of flour fit in the bin?
c) How many kilograms of flour does the bin hold?
d) The bakery stores salt in a bin that has one half the height and one half the diameter. Using your answer from part a) and proportional reasoning for volume, what is the volume of salt in the bin?
4. A hot water tank consists of a heavy steel tank, a layer of insulation, and an outer shell. The outer dimensions of a cylindrical hot water tank are 62.2 cm diameter and 149.9 cm height, and the rated water capacity is 270 L.
a) What is the total volume, in litres, of the hot water tank?
b) Compare the total volume of the hot water tank to its rated capacity, expressed as a percent.
Many people have a hot water tank in their homes, which they use to heat water for everyday tasks such as cooking, bathing, and space heating.
5. A machine shop ships a machined steel shaft section to a customer by courier. To protect the surface finish, they wrap the shaft in a layer of bubble wrap plastic and put it in a cardboard box.
The shaft has a diameter of 40 mm and a length of 475 mm. The bubble wrap is 10 mm thick. The box is 6 cm square by 50 cm long.
a) What area of bubble wrap (in cm2) is needed? (The ends of the shaft are not covered.)
b) How much additional packing material is needed to fill the box?
MathWorks 11 146
6. Plastic extrusion is an efficient method of manufacturing complex shapes.
Melted plastic is squeezed through a shaped die or nozzle into long sections whose cross-sections have the desired shape.
A plastic extrusion has the cross-section shown in the diagram. The outer pipe has three internal channels for separate flows. What volume of plastic is needed per metre (1000 mm) of extrusion?
6 mm
3 mm 3 mm
20 mm
extend your thinking
7. What is the volume of the cylindrical section shown in the diagram?
a = 218°
r = 0.47 m
h = 1.5 m
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PRojeCt—DeSIGnInG PACKAGeS desIgn your paCkagIng
Now that you have researched the dimensions commonly used for packaging your item, it is time to put together your own version. You can start by producing a professional-looking sketch of the item you are designing the packaging for, labelled with its measurements.
Next, make sketches of three versions of packaging for your item. As you design, consider cost and eye appeal. Your package should be relatively inexpensive to produce, but its shape and colour should also be appealing enough to attract your customer.
• Determine the measurements of your three packages and label them with these values.
• Determine and record the length, height, width, and surface area of your three designs.
When you have completed these steps, you can use your three designs to make materials for your presentation. Materials can include a poster that displays your designs, handouts, or an electronic presentation.
After you draw your packaging designs, you will need to label each one with its dimensions and determine its surface area.
puzzle It out Cube
Sarah has created a cube where the numerical value of the surface area is equal to the numerical value of the volume. What is the side length of the cube?