SmarterEd

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Volume, SMB-001

A skip bin is in the shape of a trapezoidal prism, with dimensions as shown.
 

Find the volume of the skip bin in cubic metres.   (3 marks)

Show Answers Only

`5.4\ text(m)^3`

Show Worked Solution
`text(Area of trapezoid)` `= 1/2h (a + b)`
  `= 1/2 xx 1.2 xx (3.6 + 2.4)`
  `= 3.6\ text(m)^2`

 

`:.\ text(Volume)` `= Ah`
  `= 3.6 xx 1.5`
  `= 5.4\ text(m)^3`

Area, SMB-015

A letterbox, which has been constructed by joining a half cylinder and rectangular prism, needs to be painted on each external surface.
 

Find the surface area of the letterbox, giving your answer correct to the nearest square centimetre.   (3 marks)

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`9061\ text(cm)^2`

Show Worked Solution
`text{Surface Area (half cylinder)}` `= \frac{1}{2} (2pi r^2 + 2pi rh)`
  `= \frac{1}{2} (2pi xx 17.5^2 + 2pi xx 17.5 xx 40)`
  `= 3161.227…\ text{cm}^2`

 

`text{Area (rest)}` `= (35 xx 40)+2(35 xx 30) + 2(40 xx 30)`  
  `=1400+2(1050)+2(1200)`  
  `=5900\ text{cm}^2`  

 
`text{SA (total)}\ = 3161 + 5900 = 9061\ text{cm}^2`

Area, SMB-014

Calculate the surface area of the half-cylinder below (including the base), giving your answer correct to two decimal places.   (3 marks)
 

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`848.29\ text(m)^2`

Show Worked Solution
`text(Surface Area)` `= \frac{1}{2} (2pi r^2 + 2pi rh) + (7 xx 45)`
  `= \frac{1}{2} (2pi xx 3.5^2 + 2pi xx 3.5 xx 45) + 315`
  `= 848.285…`
  `= 848.29\ \text{m}^2\ \ text{(2 d.p.)}`

Area, SMB-013

Calculate the surface area of the cylinder below, giving your answer correct to two decimal places.   (2 marks)
 

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`58.31\ text(m)^2`

Show Worked Solution
`text(Surface Area)` `= 2pi r^2 + 2pi rh`
  `= 2pi (1.6)^2 + 2 pi xx 1.6 xx 4.2`
  `= 58.307…`
  `= 58.31\ \text{m}^2\ \ text{(2 d.p.)}`

Area, SMB-035

The surface area of a brick is given by the rule:

total surface area = 2 × [(width × height) + (width × length) + (height × length)]

The brick shown has a total surface area of 982 square centimetres.
 

What is the width of the brick?   (2 marks)

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`10.5\ text(cm)`

Show Worked Solution

`text(Let)\ \ w = text(width of the brick)`

`text(Surface Area)` `= 2 xx [8w + 22w + (8 xx 22)]`
`982` `= 2(30w + 176)`
`982` `= 60w + 352`
`60w` `= 630`
`:. w` `= 10.5\ text(cm)`

Area, SMB-034

The total surface area of a cube is 150 cm².
 

 How long is an edge of the cube?   (2 marks)

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`text(5 cm)`

Show Worked Solution

`text(Surface area of 1 face of the cube)`

`=150/6`

`=25\ text(cm)^2`
 

`text(Let)\ \ x=\ text(length of 1 side)`

`x^2` `=25`
`:.\ x` `=5\ text(cm)`

Area, SMB-033

The rectangular prism, shown below, is cut in half to form 2 equal cubes.
 

What is the ratio of the surface area of the rectangular prism to one of the cubes?   (3 marks)

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`5 : 3`

Show Worked Solution

`text(S.A. of rectangular prism)`

`= 2 xx (h xx h) + 4 xx (2h xx h)`

`= 2h^2 + 8h^2`

`= 10 h^2`
 

`text(S.A. of cube)\ = 6 xx (h xx h) = 6h^2`
 

`:.\ text(Ratio of rectangular prism to cube)`

`= 10h^2 : 6h^2`

`= 5 : 3`

Area, SMB-032

A net of a cube is pictured below.
  

The cube has a volume of 512 cubic centimetres.

What is the height of the net in centimetres?   (3 marks)

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`24\ text(cm)`

Show Worked Solution
`text(Let)\ \ s` `=\ text(length of 1 side of the cube)`
`s^3` `= 512`
`s` `= 8\ text(cm)`

 

`:.\ text(Height of net)` `= 3 xx 8`
  `= 24\ text(cm)`

Area, SMB-031

A target is drawn as follows

What is the area of the entire target?

Round your answer to the nearest square centimetre.   (2 marks)

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`284\ text(cm)^2`

Show Worked Solution
`text(Area)` `= pi r^2`
  `= pi (2 + 1.5 xx 5)^2`
  `= pi (9.5)^2`
  `= 283.52…`
  `= 284\ text(cm)^2`

Area, SMB-029 MC

The plan of a square area of land is drawn below.

A mining licence is granted over part of the land.

All measurements are given in kilometres.
 

The area of land the mining licence covers is

  1. `x^2-20`
  2. `x^2 + 20`
  3. `2x^2-9`
  4. `2x^2-20`
Show Answers Only

`A`

Show Worked Solution

`text{Area of original land}\ = x xx x = x^2\ text{km}^2`

`text{Size of non-mining area}\ = 4 xx 5 = 20\ text{km}^2`

`:.\ text(Area of mining)\ = (x^2-20)\ text{km}^2`

`=>A`

Area, SMB-027

Kate owns a farm with a square paddock. She increases the area of the paddock by changing it into the shape of trapezium.
 

What is the area of the new paddock, in square metres?   (2 marks)

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`1505\ text(m²)`

Show Worked Solution
`A` `= 1/2h(a + b)`
  `= 1/2 xx 35 xx (35 + 51)`
  `= 1505\ text(m²)`

Area, SMB-026

Clancy needs to paint all sides of a triangular prism.

The area of each triangular face is 12 cm².

What is the total area Clancy needs to paint?   (2 marks)

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`78\ text(cm²)`

Show Worked Solution
`text(Total Area)` `= (2 xx 12) + 2 xx (3 xx 5) + (8 xx 3)`
  `= 78\ text(cm²)`

Area, SMB-025

An irregular shaped lawn is pictured below, with all measurements in metres.
  

 

What is the area of the lawn in square metres?   (3 marks)

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`65\ \text{m}^2`

Show Worked Solution

`text(Split the lawn into 3 areas)`

`text(Area 1) = 9 xx 3 = 27\ text(m²)`

`text(Area 2) = 5 xx 2 = 10\ text(m²)`

`text(Area 3:)`

`text(Height) = 5+2 = 7\ text(m)`

`text(Width)= 12-(5 + 3)= 4\ text(m)`

`=> text(Area 3) = 7 xx 4 = 28\ text(m²)`

 

`:.\ text(Area of lawn)` `= 27 + 10 + 28`
  `= 65\ text(m²)`

Area, SMB-024

A plan of Zev's backyard is shown.
  

 

Zev wants to fertilise the lawn in his backyard and needs to know the total area.

What is the total area of Zev's lawn, in square metres?   (2 marks)

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`text(26 m²)`

Show Worked Solution

`text(Area)` `=\ text(square + triangle)`
  `= (4 xx 4) + (1/2 xx 4 xx 5)`
  `= 16 + 10`
  `= 26\ text(m²)`

Area, SMB-023 MC

Harry has an irregularly shaped back yard.
  

What is the area of Harry's back yard?

  1. `text(93 m)²`
  2. `text(384 m)²`
  3. `text(544 m)²`
  4. `text(768 m)²`
Show Answers Only

`B`

Show Worked Solution
`text(Area)` `= (1/2 xx 14 xx 32) + (1/2 xx 10 xx 32)`
  `= 224 + 160`
  `= 384\ text(m)²`

 
`=>B`

Area, SMB-022

A cube has a side length of 6 cm.

Two smaller cubes of side length 3 cm are attached to the larger cube as shown in the diagram below.
  

What is the surface area of the new object?   (3 marks)

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`270\ text(cm²)`

Show Worked Solution

`text(One strategy:)`

`text{Calculate the surface area (S.A.) of the 2 objects joined}`

`text(separately and deduct the area where they join.)`
 

`text{S.A. (large cube)} = 6 xx 6xx 6 = 216\ text(cm²)`

`text{S.A. (2 smaller cubes joined together)}`

`=2 xx (3 xx 3) + 4 xx (6 xx 3)`

`=90\ text(cm²)`

`text{S.A. to be deducted (where cubes are attached)}`

`= 2 xx (6 xx 3)`

`= 36\ text(cm²)`

 

`:.\ text{S.A. (new object)}` `= 216 + 90-36`
  `=270\ text(cm²)`

Area, SMB-021 MC

Pablo cuts a square out of a rectangular piece of paper, as shown below.

The square Pablo cut out has a side length of 6 cm.

Which of these expressions gives the area of Pablo's piece of paper after cutting out the square?

  1. `(28 xx 20)-(6 xx 6)`
  2. `(22 xx 14) + (6 xx 6)`
  3. `(28 + 20)-(6 + 6)`
  4. `(28 + 20) + (6 xx 6)`
Show Answers Only

`A`

Show Worked Solution
`text(Area)` `=\ text(Area of larger rectangle − Area of square)`
  `= (28 xx 20)-(6 xx 6)`

 
`=>A`

Area, SMB-020

A trapezium is constructed on a grid of 10 rectangles.

Each rectangle measures  3 cm × 7 cm.

 

Find the area of the trapezium?   (2 marks)

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`168\ text(cm²)`

Show Worked Solution
`text(Area 1 rectangle)` `= 3 xx 7`
  `= 21\ text(cm²)`

 

`:.\ text(Total Area)` `= (6 xx 21) + 2\ text(triangles)`
  `= 6 xx 21 + 2 xx (1/2 xx 3 xx 14)`
  `= 126 + 42`
  `= 168\ text(cm²)`

Area, SMB-019 MC

Olive drew this plan of her lawn.

Which expression gives the area of Olive's lawn?

  1. `(a xx b) + (c xx d)`
  2. `(a xx b) xx (c xx d)`
  3. `(a + b) + (c + d)`
  4. `(a + b) xx (c + d)`
Show Answers Only

`A`

Show Worked Solution

`text(Total Area)` `= text(Area 1) + text(Area 2)`
  `= (a xx b) + (c xx d)`

`=> A`

Area, SMB-012

Find the surface area of the solid below where all measurements are in metres.  (2 marks)
 

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\(104\ \text{m}^2\)

Show Worked Solution

\(\text{Area (front side)} = (4 \times 2) + (2 \times 2) = 12\ \text{m}^2 \)

\(\text{S.A.}\) \(=2 \times 12 + 2 \times (5 \times 4) + 4 \times (5 \times 2) \)  
  \(= 24 + 2(20) + 4(10) \)  
  \(=104\ \text{m}^2\)  

Area, SMB-011

A triangular prism has the following dimensions (all measurements are in metres).
 

  1. Draw a net of the solid   (1 mark)
  1. Find the surface area of the prism.  (2 marks)
Show Answers Only

i. 

ii.    \(348\ \text{m}^2\)

Show Worked Solution

i.    

ii.    \(\text{S.A.}\) \(=2 \times (5 \times 18) + (8 \times 18) + 2 \times (\dfrac{1}{2} \times 8 \times 3) \)  
  \(= 2(90) + 144 + 2(12) \)  
  \(=348\ \text{m}^2\)  

Area, SMB-010

A triangular prism has the following dimensions (all measurements are in centimetres).
 

  1. Draw a net for this solid.   (1 mark)
  1. Find the surface area of the prism.  (2 marks)
Show Answers Only

i.  

 

ii.    \(408\ \text{cm}^2\)

Show Worked Solution

i.   

ii.    \(\text{S.A.}\) \(=(15 \times 6) + (15 \times 10) + (15 \times 8) + 2 \times (\dfrac{1}{2} \times 8 \times 6) \)  
  \(=90+150+120+2(24) \)  
  \(=408\ \text{cm}^2\)  

Area, SMB-009

Find the surface area of the rectangular prism pictured below, correct to 1 decimal place. All measurements are in metres.  (2 marks)
 

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\(68.0\ \text{m}^2\)

Show Worked Solution
\(\text{S.A.}\) \(=2 \times (2.8 \times 1.7) + 2 \times (2.8 \times 6.5) + 2 \times (1.7 \times 6.5) \)  
  \(= 2(4.76 + 18.2 + 11.05) \)  
  \(=68.02\)  
  \(=68.0\ \text{m}^2\ \ \text{(1 d.p.)}\)  

Area, SMB-008

Find the surface area of the rectangular prism pictured below. All measurements are in metres.  (2 marks)
 

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\(368\ \text{m}^2\)

Show Worked Solution
\(\text{S.A.}\) \(=2 \times (6 \times 14) + 2 \times (6 \times 5) + 2 \times (5 \times 14) \)  
  \(=2(84+30+70)\)  
  \(=368\ \text{m}^2\)  

Area, SMB-007

The diagram shows the shape of Carmel’s garden bed. All measurements are in metres.
  

Show that the area of the garden bed is 57 square metres.  (2 marks)

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`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution
`text(Area of)\ Delta ABC` `= 1/2 xx b xx h`
  `= 1/2 xx 10 xx 5.1`
  `= 25.5\ text(m²)`
`text(Area of)\ Delta ACD` `= 1/2 xx 10 xx 6.3`
  `= 31.5\ text(m²)`

 

`:.\ text(Total Area)` `= 25.5 + 31.5`
  `= 57\ text(m² … as required)`

Area, SMB-001

Find the area of the shaded part in the diagram below, giving your answer in square metres to 1 decimal place.  (3 marks)
 

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\(62.8\ \text{m}^2 \)

Show Worked Solution
\(\text{Shaded area}\) \(=\ \text{Area of large sector – Area of small sector}\)  
  \(= \dfrac{30}{360} \times \pi R^2-\dfrac{30}{360} \times \pi r^2 \)  
  \(= \dfrac{1}{12} \pi (16^2-4^2) \)  
  \(= 62.83… \)  
  \(=62.8\ \text{m}^2\ \ \text{(1 d.p.)} \)  

Area, SMB-008 MC

Four identical circles of radius `r` are drawn inside a square, as shown in the diagram below

The region enclosed by the circles has been shaded in the diagram.
 

 
The shaded area can be found using

  1. `4 r^2-2 pi r`
  2. `4 r^2-pi r^2`
  3. `4 r-pi r^2`
  4. `2 r^2-pi r^2`
Show Answers Only

`B`

Show Worked Solution

`text(Area of square) = 4r xx 4r = 16r^2`

`text(Area of circles) = 4 pi r^2`

`text(Shaded Area)`

`= 1/4 xx (16r^2-4 pi r^2)`

`= 4 r^2-pi r^2`
 

`=>  B`

Area, SMB-006

A child's toy has the following design.
 

Find the area of the shaded region to the nearest square centimetre.  (3 marks)

Show Answers Only

`27\ text(cm²)`

Show Worked Solution

`text{Circle radius = 3 cm}`

`text(Consider the rectangle starting from the middle of the left circle.)`

`text{Shaded Area}` `= text{Area rectangle}-3.5 xx text{Area circle}`
  `= (21 xx 6)-3.5 xx pi xx 3^2`
  `= 27.03 …`
  `=27\ text{cm²  (nearest whole)}`

Area, SMB-005

A path 1.5  metres wide surrounds a circular lawn of radius 3 metres. 
  

Find the area of the path, correct to the nearest square metre.   (2 marks)

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`35\ text{m²}`

Show Worked Solution

`text(Area of annulus)`

`= pi (R^2-r^2)`

`= pi (4.5^2-3^2)`

`= pi (11.25)`

`=35.3…`

`=35\ text{m²  (nearest m²)}`

Area, SMB-004

The diagram shows the floor of a shower. The drain in the floor is a circle with a diameter of 10 cm.

Find the area of the shower floor, excluding the drain, to the nearest square centimetre.   (3 marks)
 

Show Answers Only

`9921\ text(cm²)`

Show Worked Solution
`text(Area)` `=\ text(Square – Circle)`
  `= (100 xx 100)-(pi xx 5^2)`
  `= 10\ 000-78.5398…`
  `= 9921.46…\ text(cm²)`
  `= 9921\ text{cm² (nearest cm²)}`

Area, SMB-003

The shaded region shows a quadrant with a rectangle removed.
  

Find the area of the shaded region, to the nearest cm2.   (2 marks)

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`52\ text(cm²)`

Show Worked Solution
`text(Shaded area)` `=\ text(Area of segment – Area of rectangle)`
  `=1/4 pi r^2-(6xx2)`
  `=1/4 pi xx9^2-12`
  `=51.617…`
  `=52\ text(cm²)`

Area, SMB-002

 What is the area of the shaded part of this quadrant, to the nearest square centimetre?  (3 marks)

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`42\ text{cm²}`

Show Worked Solution
`text(Area)` `=\ text(Area of Sector – Area of triangle)`
  `= (theta/360 xx pi r^2)-(1/2 xx l xx h)`
  `= (90/360 xx pi xx 8^2)-(1/2 xx 4 xx 4)`
  `= 50.2654…-8`
  `= 42.265…\ text(cm²)`
  `= 42\ text{cm²  (nearest cm²)}`

Trigonometry, SMB-031

  1. Express the ratio of `tan theta`  as a fraction.  (2 marks)

  2. Hence or otherwise, find the value of `theta`, correct to two decimal places.  (1 mark)

Show Answers Only
  1. `tan theta=15/8`
  2. `61.93^@`
Show Worked Solution

i.    `text{By Pythagoras,}`

`x^2+8^2` `=17^2`  
`x^2` `=289-64`  
  `=225`  
`x` `=15`  

 
`:.tan theta = 15/8`

 

ii.   `theta` `=tan^{-1}(15/8)`  
  `=61.927…^@`  
  `=61.93^@\ text{(to 2 d.p.)}`  

Trigonometry, SMB-030

Find the value of `theta`, correct to the nearest minute.  (2 marks)

Show Answers Only

`54^@ 28^{′}`

Show Worked Solution
`tan theta` `=7/5`  
`theta` `=tan^{-1}(7/5)`  
  `=54.462…`  
  `=54^@ 27^{′}44^{″}`  
  `=54^@ 28^{′}`  
NOTE: Seconds > 30 are rounded up to the higher minute.

Trigonometry, SMB-029

Find the value of `theta`, correct to the nearest minute.  (2 marks)

Show Answers Only

`61^@ 56^{′}`

Show Worked Solution
`tan theta` `=15/8`  
`theta` `=tan^{-1}(15/8)`  
  `=61.927…`  
  `=61^@55^{′}39^{″}`  
  `=61^@ 56^{′}`  
NOTE: Seconds > 30 round up to the higher minute.

Trigonometry, SMB-024

Express `tan theta` as a fraction in its simplest form.  (3 marks)

Show Answers Only

`tan theta = (2sqrt5)/5`

Show Worked Solution

`text{Let unknown side =}\ x`

`text{By Pythagoras:}`

`6^2` `=x^2 + 4^2`  
`x^2` `=36-16`  
`x` `=sqrt20\ \ (x>0)`  

 

`:.tan theta` `=4/sqrt20`  
  `= 4/(2sqrt5) xx sqrt5/sqrt5`  
  `= (2sqrt5)/5`  

Trigonometry, SMB-023

Using Pythagoras and showing your working, express `tan 30^{@}` as a fraction.  (2 marks)

Show Answers Only

`tan 30^{@} = 1/sqrt3`

Show Worked Solution

`text{Let}\ \ x =\ text{unknown side}`

`text{By Pythagoras:}`

`2^2` `=x^2 + 1^2`  
`x^2` `=4-1`  
`x` `=sqrt3\ \ (x>0)`  

 
`tan 30^{@} = text{opp}/text{adj} = 1/sqrt3`

Trigonometry, SMB-022

Find the value of `theta`, correct to the nearest minute.  (3 marks)

Show Answers Only

`66^@ 25^{′}`

Show Worked Solution

`cos theta` `=2/5`  
`theta` `=cos^{-1}(2/5)`  
  `=66.421…^@`  
  `=66^@ 25^{′} text{(to 1 d.p.)}`  

Trigonometry, SMB-021

  1. Express the ratio of `cos theta`  in its simplest form.  (2 marks)

  2. Hence, find the value of `theta`, correct to two decimal places.  (1 mark)

Show Answers Only
  1. `cos theta = 4/5`
  2. `theta = 36.87^@`
Show Worked Solution

i.    `text{By Pythagoras,}`

`x^2+8^2` `=10^2`  
`x^2` `=100-36`  
  `=64`  
`x` `=8`  

 
`:.cos theta = 8/10=4/5`

 

ii.   `theta` `=cos^{-1}(4/5)`  
  `=36.869…^@`  
  `=36.87^@\ text{(to 2 d.p.)}`  

Trigonometry, SMB-020


  

Find the value of `theta`, correct to the nearest minute.  (2 marks)

Show Answers Only

`70^@ 32^{′}`

Show Worked Solution
`cos theta` `=4/12`  
`theta` `=cos^{-1}(1/3)`  
  `=70.528…`  
  `=70^@ 31^{′}43^{″}`  
  `=70^@ 32^{′}`  
NOTE: Seconds > 30 are rounded up to the higher minute.

Trigonometry, SMB-014

Express `cos theta` as a fraction in its simplest form.  (3 marks)

Show Answers Only

`cos theta = sqrt3/2`

Show Worked Solution

`text{Let unknown side =}\ x`

`text{By Pythagoras:}`

`4^2` `=x^2 + 2^2`  
`x^2` `=16-4`  
`x` `=sqrt12\ \ (x>0)`  

 

`:.cos theta` `=sqrt12/4`  
  `= (2sqrt3)/4`  
  `= sqrt3/2`  

Trigonometry, SMB-013

Using Pythagoras and showing your working, express `cos 30^{@}` as a fraction.  (2 marks)

Show Answers Only

`cos 30^{@} = sqrt3/2`

Show Worked Solution

`text{Let}\ \ x =\ text{unknown side}`

`text{By Pythagoras:}`

`2^2` `=x^2 + 1^2`  
`x^2` `=4-1`  
`x` `=sqrt3\ \ (x>0)`  

 
`cos 30^{@} = text{adj}/text{hyp} = sqrt3/2`