MATH on the Job
Proper safety gear, such as goggles and gloves, is required when installing drywall.
Shay is a drywall installer who works in Iqaluit, Nunavut. To become a drywall installer, Shay completed the Trades Access program at Aurora College in Yellowknife, NT. Two months later, he was employed in a full-time job. At work, Shay is responsible for measuring and cutting drywall, laying out reference lines and points, and reading blueprints to determine the best way to install drywall.
When he started working, Shay earned $20.00 an hour. As a drywall installer, Shay needs to be in good physical condition and be aware of proper lifting techniques because he has to lift large drywall sheets. When drywall is cut, it produces gypsum dust, so avoiding exposure to this dust is also important for Shay’s health and safety.
At work, Shay applies plasterboard or other wallboard to ceilings and interior walls of buildings. A standard sheet of drywall measures 4 feet by 8 feet, with the 8-foot length installed horizontally. For his next job, he has been asked to cover one wall in each of four different rooms.
The dimensions of each wall are
• Wall 1: 14 feet wide by 12 feet high
• Wall 2: 16 feet wide by 12 feet high
• Wall 3: 10 feet wide by 12 feet high
• Wall 4: 20 feet wide by 12 feet high
1. How many sheets will Shay need to cover each wall?
2. What is the minimum number of sheets Shay will need to cover all of the walls?
explore the Math
Have a look around you. How many objects can you see that can be described by only two dimensions (for example, length and width)? Flat surfaces can be described this way, such as the surface of a desk, the size of a window, and the cover of a book. However, we live in a three-dimensional world, and almost all objects need a third dimension to describe them. For example, a box has length, width, and height.
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boxes are commonly constructed in the shape of a prism.
Imagine a pizza box: it is a rectangular prism. Each of its sides is a flat surface whose area can be calculated using length and width.
rectangular prism:
a three-dimensional shape with ends that are congruent rectangles and with sides that are parallelograms The box can be made from a single sheet of cardboard; you
assemble the box by first dividing the cardboard into rectangles that make up the sides of the box, and then folding the cardboard along the sides of the rectangles. This flat representation of the box is called a net.
net: two-dimensional pattern that can be folded to form a three-dimensional shape The surface area of the pizza box is found by adding up the areas
of each of its sides. You can find the surface area of the box by adding up the areas of the rectangles that make up the net because the area of the sheet of cardboard is the same when it is flat as when it is folded.
surface area: the area required to cover a three-dimensional shape On flat surfaces, the units of area are squared, such as square
metres or square inches. In three dimensions, the units of surface area are also squared.
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dIsCuss the Ideas
CalCulatIng surfaCe area of a staIned glass lantern A three-dimensional object is a prism if it contains
two parallel polygons that are congruent.
prism: a three-dimensional shape with ends that are congruent polygons and with sides that are parallelograms
Congruent polygons
• have the same shape and size
• have sides and angles in the same positions 20 cm 10 cm
Each of the parallel faces is called a base, and the faces connecting the bases are called lateral faces.
base: one of the parallel faces of a prism
lateral face: a face that connects the bases of a prism
If the lateral faces are perpendicular to the bases, the prism is a right prism.
Right hexagonal prism Oblique prism
Shabina makes stained-glass garden lanterns in the shape of hexagonal prisms.
The bases (top and bottom) are hexagons and are made of metal. The faces are coloured glass.
1. Look at the lateral faces of the right hexagonal prism above. Draw a net of the right hexagonal prism.
2. If each edge of the hexagonal base is 10 cm long, and the lanterns are 20 cm high, what is the area of each piece of glass used as a lateral face?
3. What is the total surface area of glass that Shabina needs for one lantern?
Can you think of two ways to calculate the total surface area?
119 Chapter 3 Surface Area, Volume, and Capacity
Mental Math and estimation
Each shape below has been placed on a grid. If each square on the grid has a side length of 1 cm and an area of 1 cm2, estimate the area of each shape.
example 1
When constructing a fence, builders use their knowledge of angles and trigonometry.
Davinder builds a fence around his yard. To make the gate, he cuts four pieces of 2 × 4 lumber for a frame and one piece as a diagonal brace to keep the gate from sagging. For the fence pickets he uses 2 × 2 lumber, and cuts the ends at 45 degrees, as shown below.
Hinges this side 2 × 4 frame
2 × 4 brace
Garden gate
2 × 2 lumber 45° each end
Fence pickets
a) What is the shape of each of the pieces he uses for the frame?
b) What is the shape of the piece he uses for the brace?
c) What is the shape of the pickets?
solutIon
a) Each piece of the frame is in the shape of a right rectangular prism.
b) The brace is in the shape of a trapezoidal prism.
c) Each picket is in the shape of a right trapezoidal prism.
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example 2
Nicholas ships posters and reproduction art prints to customers in shipping boxes made in the shape of an equilateral triangular prism, as shown in the diagram.
a) Draw a net of the box and label the dimensions of each side.
b) Calculate the surface area of the box.
13.2 cm
96.5 cm
15.2 cm
15.2 cm
15.2 cm
solutIon
a) The net of the box would look like the following diagram.
96.5 cm
96.5 cm
15.2 cm
15.2 cm
15.2 cm
13.2 cm
b) The box is made up of three identical rectangles and two identical triangles.
Area of one rectangle A = ℓ × w
1466.8 = 96.5 × 15.2 Area of one triangle
A = 1 2
(
bh)
100.32 = 1
2(15.2× 13.2)
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Find the surface area.
Surface area = 3(1466.8) + 2(100.32) Surface area = 4400.4 + 200.64 Surface area = 4601.04
Since all the measurements have one decimal, the answer should be rounded to one decimal place. The surface area of the box is 4601.0 cm2.
ACtIVIty 3.1 hexoMInos
A hexomino is a shape made of six identical squares connected along their sides. There are 35 different patterns that can be made from six squares to create 35 hexominos. Below are five different hexominos.
Work with a team to complete the activity.
1. Which of these hexominos can be folded to form a closed cube?
2. Create the remaining 30 hexominos. Identify which hexominos will create a closed cube. Be prepared to defend your choices to the rest of your class.
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dIsCuss the Ideas
surfaCe area of CabInets
1. Karl is building a set of cabinets. He makes the first one 40 cm long × 40 cm deep × 70 cm high. When the cabinet is fully closed, it is the shape of a rectangular prism. What is the surface area of the cabinet?
2. Karl makes the second cabinet 2 times as long, but with the same depth and height. What is the surface area of the second cabinet?
3. Karl makes the third cabinet 2 times as long and 2 times as high as the first one, but with the same depth. What is the surface area of the third cabinet?
4. Examine the following table showing how the surface area changes when the dimensions are changed.
surfaCe area ratIos
Cabinet Length Depth Height Surface
area (Surface Area)
(SurfaceArea of Cabinet 1)
1 L D H 14 400 1.00
2 2L D H 23 200 1.61
3 2L D 2H 40 000 2.78
4 2L 2D 2H ? ?
What do you notice about the surface area ratio when the length is doubled? How do the surface areas compare when length and height are doubled?
5. Calculate the surface area of a cabinet where all three dimensions are double those of cabinet one, and calculate the ratio of surface areas. What can you conclude about the relationship between the scale factor used to create the new dimensions and the ratio of the surface areas?
example 3
A simple child’s wagon is constructed out of wood and a t-handle bar.
Katie is building a child’s wagon. The wagon box is to be 3 feet long and 18 inches wide. The sides of the wagon will be 10 inches high.
a) Draw a net of the wagon box and label the dimensions.
b) Calculate the surface area of the box, in square inches.
c) One square foot equals 144 in2. What is the surface area of the box in square feet?
d) Can she make the box from a single sheet of 4 feet × 8 feet plywood?
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solutIon
a)
36 in
10 in 18 in
b) Find the surface area using this formula.
SA = area of base of wagon + area of ends + area of sides SA = (36 × 18) + 2(36 × 10) + 2(18 × 10)
SA = 1728 in2
The surface area of the box is 1728 in2.
c) To find the number of square feet, divide the number of square inches by 144.
1728 ÷ 144 = 12
The surface area of the box is 1728 in2, which is equal to 12 ft2. d) Calculate the number of square feet in a sheet of plywood.
4 × 8 = 32 ft2
Since there are 32 square feet of plywood, it looks like the wagon box can be made from a single sheet of plywood. Further calculations need to be done, though, to make sure that the dimensions of the net will fit on one sheet.
(length of the base + 2 side heights) = 36 + (2 × 10) (length of the base + 2 side heights) = 56
56 inches is less than 8 feet.
(width of the base + 2 side heights) = 18 + (2 × 10) (width of the base + 2 side heights) = 38
38 inches is less than 4 feet.
Therefore, the net will fit on a single sheet of plywood.
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ACtIVIty 3.2
buIldIng a shIppIng Crate
Shipping crates come in different sizes.
Gerald is building a shipping crate. Each crate is a right rectangular prism, and needs to hold 24 cubic boxes. What arrangement of boxes in the crate will result in the least amount of material required to build the crate?
1. Use a table like the one shown to figure out the different ways the 24 boxes could be arranged into a right rectangular prism. (Hint: You should find 6 boxes.) To do this
• determine the surface area for each resulting right rectangular prism;
and
• each box should be different in shape, meaning do not create a box that can be turned to look like another one of your boxes.
Crate dIMensIons
Length Width height Surface area
sa mp le
2. Which box will require the least amount of material to create?
3. If each cubic box has a side length of 10 ft, what is the surface area of the box you chose above?
buIld your skIlls
1. Darcy has a summer job painting houses. He is asked to paint the wooden siding on a house that is 28 feet wide and 35 feet long. The siding extends 6 feet up the side of the house.
a) What is the total surface area that he must paint? (Ignore the area of windows, doors, and stairs.)
b) A one-gallon can of the stain that Darcy is using covers approximately 225 ft2. If Darcy applies 2 coats of stain, how many cans of stain should he buy?
125 Chapter 3 Surface Area, Volume, and Capacity
2. Maillardville is one of the oldest French-speaking settlements in British Columbia. The town’s Festival du Bois celebrates francophone culture.
Festival-goers enjoy francophone-inspired music, dance, and treats.
Maillardville, bC hosts the annual Festival du Bois.
the festival celebrates francophone music, food, and culture.
Mireille builds wooden dividers, which are placed between vendors. After the dividers are built, she must cover them in canvas.
Each rectangular divider is 3 inches wide, 80 inches high, and 60 inches long.
a) Draw the net of one divider and label its dimensions.
b) How much material will be needed to cover one divider? (Do not take into account any overlapping material.)
3. Arapoosh is a glazier. She is building a display case for a shopping mall.
The display case has a wooden hexagonal base and top. Each lateral face will be a trapezoid with a bottom width of 80 cm, a top width of 40 cm, and a height of 2 m.
a) What is the area of one lateral face of the display case?
b) What is the total area of glass she needs, in square metres?
4. Mingmei builds a shipping crate out of 14ʺ plywood. The crate is a cube with a side dimension of 3 feet.
a) What is the surface area of the crate?
b) She buys plywood in standard sheet sizes of 4 ft × 8 ft. How many sheets of plywood does she need to build one shipping crate?
c) She builds a second crate that is twice the height, but has the same length and width. How many sheets of plywood will she need to build the larger shipping crate? Explain.
5. Dirk fabricates a section of furnace duct out of sheet metal. What is the total area of sheet metal that he needs? The duct is open at the upper right face and left bottom face.
8"
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6. Zyanya looks at the floor plan of a house to estimate the area of floor tile he needs for the bathroom. He calculates that he will need approximately 64 m2 of tile. Is this a reasonable estimate? Explain. What is a possible source of Zyanya’s error? (What would be the size of a bathroom having this square footage?)
before laying tile, any old material must be removed and the surface of the floor must be made clean and level.
extend your thinking
7. Wolfgang wants to paint the roof of his workshop, which has a corrugated steel roof as shown in the diagram below.
The length of the roof is 25 feet, and the width is 20 feet. He calculates the area of the roof using the following dimensions:
25’
20’
4”
6”
Area = length × width Area = 25 × 20 Area = 500 ft2
He buys enough paint to cover 500 ft2. Just over halfway through the job, he runs out of paint. Explain why this happened. What is the area that he should have used to calculate the amount of paint?