SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

v1 Measurement, STD1 M5 2021 HSC 26

The diagrams show two similar shapes. The dimensions of the small shape are enlarged by a scale factor of 1.5 to produce the large shape.
 

Calculate the area of the large shape.  (3 marks)

Show Answers Only

`94.5\ text(cm)^2`

Show Worked Solution

`text(Dimension of larger shape:)`

♦♦ Mean mark 32%.

`text(Width) = 6 xx 1.5 = 9\ text(cm)`

`text(Height) = 8 xx 1.5 = 12 \ text(cm)`

`text(Triangle height) = 2 xx 1.5 = 3\ text(cm)`

`:.\ text(Area)` `= 9 xx (12-3) + 1/2 xx 9 xx 3`
  `= 94.5\ text(cm)^2`

v1 Measurement, STD2 M1 2022 HSC 34

A composite solid is shown. The top section is a hemisphere with a diameter of 6 cm. The bottom section is a cylinder with a height of 3 cm and a diameter of 4 cm
 

Find the total volume of the composite solid in cm³, correct to 1 decimal place.  (4 marks)

Show Answers Only

`94.2 \ text{cm}^3`

Show Worked Solution
`text{Volume of Hemisphere}` `=2/3 pi r^3`  
  `=2/3 pi xx 3^3`  
  `=56.54\ text{cm}^3`  

 

`text{Volume of Cylinder}` `=pi r^2 h`  
  `=pi (2^2) xx 3`  
  `=37.69\ text{cm}^3`  

 

`text{Total Volume}` `=56.54+37.69`
  `=94.23`
  `=94.2 \ text{cm}^3`

v1 Measurement, STD2 M1 2013 HSC 12 MC

A hemisphere sits perfectly on top of a cylinder to form a solid. 

What is the volume of the solid?

  1. 1750 cm³
  2. 1950 cm³
  3. 2150 cm³
  4. 2350 cm³
Show Answers Only

`C`

Show Worked Solution
`text(Volume )` `=text{Vol (cylinder)} +text{Vol (hemisphere)}`
  `= pi r^2h+2/3pi r^3`
  `= pi xx 6^2 xx 15 + 2/3pi xx 6^3`
  `=2149.84\ text(cm)^3`

 
`=>\ C`

v1 Measurement, STD2 M1 2021 HSC 16

The surface area, `A`, of a sphere is given by the formula

`A = 4 pi r^2,`

where `r` is the radius of the sphere.

A satellite dish resembles the inner surface of the lower half of a sphere with a radius of 1.5 meters.

 

Find the surface area of the satellite dish in square metres, correct to one decimal place.   (2 marks)

Show Answers Only

`14.1\ text{m}^2`

Show Worked Solution
`A` `= frac{1}{2} times 4 pi r^2`
  `= 2 pi r^2`
  `= 2 pi times (1.5)^2`
  `= 2 pi times 2.25`
  `= 14.137…`
  `= 14.1\ text{m}^2\ \text{(1 d.p.)}`

v1 Measurement, STD2 M1 2019 HSC 16

A decorative light fixture is in the shape of a hollow hemisphere with a diameter of 24 cm.
 

The inside of the fixture is to be coated with reflective paint.

What is the area to be painted on the inside surface? Give your answer correct to the nearest square centimetre.   (2 marks)

Show Answers Only

`905\ \text{cm}^2`

Show Worked Solution
`A` `= 2 pi r^2`
  `= 2 × pi × 12^2`
  `= 2 × pi × 144 = 905.0…`
  `≈ 905\ \text{cm}^2`

Probability, SMB-014

Students studying vocational education courses were surveyed about their living arrangements.
  

  1. One of these students is selected at random. What is the probability, correct to the nearest percentage, that this student is male and living with his parent(s)?   (2 marks)

  2. A female student is selected. What is the probability that she is not living with her parent(s)?   (2 marks)

Show Answers Only
  1.  `\text{31%}`
  2. `91/114`
Show Worked Solution

i.     `text{Number of males living with parents = 155}`

`text{Total students surveyed = 505}`

`P\text{(male and living with parents)}` `=155/505`  
  `=0.3069…`  
  `=31\text{%  (nearest %)}`  

 
ii.    `text{Number of females = 228}`

`text{Females not living with parents = 182}`

`P\text{(selected female not living with parents)} = 182/228 = 91/114`

Probability, SMB-013

A group of coalminers were surveyed about what registered vehicles they own.

They were surveyed on whether they own a car, a motorbike, both or neither and the results were recorded in the Venn diagram below.
 

Record this information in the partially completed 2-way table below.    (2 marks)

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Car}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Car}\\
\hline
\rule{0pt}{2.5ex}\text{Motorbike}\rule[-1ex]{0pt}{0pt} &  &  8 \\
\hline
\rule{0pt}{2.5ex}\text{No Motorbike}\rule[-1ex]{0pt}{0pt} &  &  \\
\hline
\end{array}

Show Answers Only

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Car}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Car}\\
\hline
\rule{0pt}{2.5ex}\text{Motorbike}\rule[-1ex]{0pt}{0pt} & 7 &  8 \\
\hline
\rule{0pt}{2.5ex}\text{No Motorbike}\rule[-1ex]{0pt}{0pt} & 29 & 6 \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Car}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Car}\\
\hline
\rule{0pt}{2.5ex}\text{Motorbike}\rule[-1ex]{0pt}{0pt} & 7 &  8 \\
\hline
\rule{0pt}{2.5ex}\text{No Motorbike}\rule[-1ex]{0pt}{0pt} & 29 & 6 \\
\hline
\end{array}

Probability, SMB-012

A group of 20 museum visitors were surveyed about what languages they could speak fluently.

They were surveyed on whether they could speak English or French and the results were recorded in the Venn diagram below.
 

Record this information in the partially completed 2-way table below.    (2 marks)

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} & \text{English}\rule[-1ex]{0pt}{0pt} & \text{No English}\\
\hline
\rule{0pt}{2.5ex}\text{French}\rule[-1ex]{0pt}{0pt} &  &  \\
\hline
\rule{0pt}{2.5ex}\text{No French}\rule[-1ex]{0pt}{0pt} & 8 &  \\
\hline
\end{array}

Show Answers Only

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} & \text{English}\rule[-1ex]{0pt}{0pt} & \text{No English}\\
\hline
\rule{0pt}{2.5ex}\text{French}\rule[-1ex]{0pt}{0pt} & 5 & 3 \\
\hline
\rule{0pt}{2.5ex}\text{No French}\rule[-1ex]{0pt}{0pt} & 8 & 4 \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} & \text{English}\rule[-1ex]{0pt}{0pt} & \text{No English}\\
\hline
\rule{0pt}{2.5ex}\text{French}\rule[-1ex]{0pt}{0pt} & 5 & 3 \\
\hline
\rule{0pt}{2.5ex}\text{No French}\rule[-1ex]{0pt}{0pt} & 8 & 4 \\
\hline
\end{array}

Probability, SMB-011

A class of 30 students were surveyed about their pets. They were asked whether they owned a dog, cat, both or neither and the results were recorded in the Venn diagram below.
 

Record this information in the partially completed 2-way table below.    (2 marks)

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Dog}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Dog}\\
\hline
\rule{0pt}{2.5ex}\text{Cat}\rule[-1ex]{0pt}{0pt} &  &  \\
\hline
\rule{0pt}{2.5ex}\text{No Cat}\rule[-1ex]{0pt}{0pt} &  & 8\\
\hline
\end{array}

Show Answers Only

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Dog}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Dog}\\
\hline
\rule{0pt}{2.5ex}\text{Cat}\rule[-1ex]{0pt}{0pt} & 3 & 7 \\
\hline
\rule{0pt}{2.5ex}\text{No Cat}\rule[-1ex]{0pt}{0pt} & 12 & 8\\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \ \ \text{Dog}\ \ \ \rule[-1ex]{0pt}{0pt} & \text{No Dog}\\
\hline
\rule{0pt}{2.5ex}\text{Cat}\rule[-1ex]{0pt}{0pt} & 3 & 7 \\
\hline
\rule{0pt}{2.5ex}\text{No Cat}\rule[-1ex]{0pt}{0pt} & 12 & 8\\
\hline
\end{array}

Probability, SMB-007

Some men and women were surveyed at a football game. They were asked which team they supported. The results are shown in the two-way table.

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} &\ \textit{Team A }\ \rule[-1ex]{0pt}{0pt} &\ \textit{Team B}\ \ &\ \textit{Totals}\ \ \\
\hline
\rule{0pt}{2.5ex}\text{Men}\rule[-1ex]{0pt}{0pt} & 125 &  100 &  225 \\
\hline
\rule{0pt}{2.5ex}\text{Women}\rule[-1ex]{0pt}{0pt} & 75 & 90 & 165 \\
\hline
\rule{0pt}{2.5ex}\text{Totals}\rule[-1ex]{0pt}{0pt} & 200 & 190 & 390 \\
\hline
\end{array}

A man was chosen at random. What is the probability that he supports Team B, correct to the nearest percent?   (2 marks)

Show Answers Only

`44%`

Show Worked Solution

`text{Total number of men}\ = 225`

`text{Number of men who support Team B}\ = 100`

`P(\text{chosen man supports Team B})`

`=100/225`

`=4/9`

`=44%\ \text{(nearest %)}`

Probability, SMB-006

The subject choices in science at a high school are physics, chemistry and biology.

This Venn diagram shows the number of students who are studying each of the subjects.
 

A student studying Biology is chosen a random. 

What is the probability that the student also studies Chemistry?   (2 marks)

Show Answers Only

`7/40`

Show Worked Solution

`text(Students studying Biology)\ = 4 + 2 + 12 + 62 = 80`

`text(Students studying Biology and Chemistry)\ = 2 + 12 = 14`

`:. P\text{(chosen student studies Chemistry)}`

`= 14/80`

`=7/40`

Probability, SMB-005 MC

The subject choices in science at a high school are physics, chemistry and biology.

This Venn diagram shows the number of students who are studying each of the subjects.
 

How many of these students are studying at least two of these science subjects?

  1. `2`
  2. `24`
  3. `26`
  4. `54`
Show Answers Only

`C`

Show Worked Solution

`text(Any students in overlapping circles study 2 or 3)`

`text(of the science subjects.)`

`:.\ text(Number of students)\ = 8 + 12 + 4 + 2 = 26`

`=>C`

Probability, SMB-004

Zilda took a survey of eighteen year olds, asking if they work, go to school, do both or do neither.

The Venn diagram shows the results.
 

 

What is the probability that a person randomly selected from the group goes to school and works, rounded to three decimal places?   (2 marks)

Show Answers Only

`0.067`

Show Worked Solution

 `P\ text{(go to school and works)}`

`= 3/(19+3+16+7)`

`= 3/45`

`= 0.067`

Probability, SMB-003 MC

A country school surveyed 120 of its students about the type of animals they have at home.

The results are recorded in the Venn diagram below, although the number of students who only own horses is missing.
 

If one of the students is selected at random, what is the probability that the student does not own a goat?

  1. `40/120`
  2. `60/120`
  3. `80/120`
  4. `100/120`
Show Answers Only

`C`

Show Worked Solution

`text(Number of students who do not own a goat)`

`= 120-(9 + 7 + 4 + 20)`

`= 120-40`

`= 80`
 

`:.\ text(Probability) = 80/120`

`=>C`

Probability, SMB-001

A group of 125 people were asked if they wear a watch or not.

This table shows the results.
 

 

A man was selected at random.

What is the exact probability that he wears a watch?   (2 marks)

Show Answers Only

`5/12` 

Show Worked Solution

`P(text(man chosen wears watch))`

`= text(number of men wearing watch)/text(total men)`

`= 25/60`

`= 5/12`

Probability, SMB-015

On a tray there are 12 hard‑centred chocolates `(H)` and 8 soft‑centred chocolates `(S)`. Two chocolates are selected at random. A partially completed probability tree is shown.
 


 

What is the probability of selecting at least one soft-centred chocolate?  (3 marks)

Show Answers Only

`62/95`

Show Worked Solution

`P(text{at least one}\ S)`

`= 1-P(HH)`

`= 1-(12/20 xx 11/19)`

`= 1-33/95`

`= 62/95`

♦ Mean mark 45%.

Probability, SMB-014

A game consists of two tokens being drawn at random from a barrel containing 20 tokens. There are 17 red tokens and 3 black tokens. The player keeps the two tokens drawn.

  1.  Complete the probability tree by writing the missing probabilities in the boxes.  (2 marks)
     
     
       

  2.  What is the probability that a player draws at least one red token? Give your answer in exact form.  (2 marks)

Show Answers Only
  1.  
  2. `187/190`
Show Worked Solution
i.   

 

ii.   `P(text(at least one red))`

`= 1-P(BB)`

`= 1-3/20 xx 2/19`

`= 187/190`

Special Properties, SMB-025

A quadrilateral is pictured below.
 

What is the value of `x`?   (3 marks)

Show Answers Only

`126^@`

Show Worked Solution

`text{Sum of exterior angles = 360°}`

`y^{\circ}` `=360-(127+114+65)`  
  `=360-306`  
  `=54^{\circ}`  

 
`:.x^{\circ}=180-54 = 126^{\circ}\ \ \text{(180° in straight line)}`

Special Properties, SMB-026

A pentagon is pictured below, where one internal angle is a right angle.
 

What is the value of `x`?   (3 marks)

Show Answers Only

`130^@`

Show Worked Solution

`y^{\circ}=180-100 = 80^{\circ}\ \ \text{(180° in straight line)}`

`text{Sum of exterior angles = 360°}`

`z^{\circ}` `=360-(70+70+90+80)`  
  `=360-310`  
  `=50^{\circ}`  

 
`:.x^{\circ}=180-50 = 130^{\circ}\ \ \text{(180° in straight line)}`

Special Properties, SMB-024

A quadrilateral is drawn below.
 

What is the value of `x`?   (3 marks)

Show Answers Only

`117^@`

Show Worked Solution

`y^{\circ}=180-75 = 105^{\circ}\ \ \text{(180° in straight line)}`

`text{Sum of exterior angles = 360°}`

`z^{\circ}` `=360-(130+105+62)`  
  `=360-297`  
  `=63^{\circ}`  

 
`:.x^{\circ}=180-63 = 117^{\circ}\ \ \text{(180° in straight line)}`

Special Properties, SMB-030

A regular decagon is pictured below.
 

  1. What is the value of `x`?   (2 marks)

  2. What is the size of an internal angle of a decagon?   (2 marks)

Show Answers Only

i.    `36^@`

ii.   `144^@`

Show Worked Solution

i.    `text{Sum of exterior angles = 360°}`

`text{Since the decagon is regular, all external angles are equal.}`

`:.x^{\circ}= 360/10 = 36^{\circ}`

  
ii.    `text{Method 1: Using exterior angle}`

`text{Internal angle}` `=180-\text{exterior angle}`
  `=180-36`
  `=144^{\circ}`

  
`text{Method 2: Using Internal angle sum formula}`

`text{Sum of internal angles}` `=(n-2) xx 180`
  `=(10-2) xx 180`
  `=1440^{\circ}`

  
`:.\ text{Internal angle}\ = 1440/10 = 144^{\circ}`

Special Properties, SMB-024

A quadrilateral is drawn below.

What is the value of `x`?   (3 marks)

Show Answers Only

`117^@`

Show Worked Solution

`y^{\circ}=180-75 = 105^{\circ}\ \ \text{(180° in straight line)}`

`text{Sum of exterior angles = 360°}`

`z^{\circ}` `=360-(130+105+62)`  
  `=360-297`  
  `=63^{\circ}`  

 
`:.x^{\circ}=180-63 = 117^{\circ}\ \ \text{(180° in straight line)}`

Special Properties, SMB-023

A pentagon is drawn below.
 

What is the value of `x`?   (3 marks)

Show Answers Only

`99^@`

Show Worked Solution

`z^{\circ}=180-110 = 70^{\circ}\ \ \text{(180° in straight line)}`

`text{Sum of exterior angles = 360°}`

`y^{\circ}` `=360-(72+82+70+55)`  
  `=360-279`  
  `=81^{\circ}`  

 
`:.x^{\circ}=180-81 = 99^{\circ}\ \ \text{(180° in straight line)}`

Similarity, SMB-027

 


 

  1. What scale factor is used to convert Circle A into Circle B.   (1 mark)

  2. Complete this equation:
  3.      Area of Circle A = _____ × Area of Circle B.  (1 mark)

Show Answers Only
  1. \(\dfrac{1}{3}\)
  2. \(9\)
Show Worked Solution

i.     \(\text{Scale factor}\ = \dfrac{\text{Diameter B}}{\text{Diameter A}} = \dfrac{0.8}{2.4} = \dfrac{1}{3} \)
 

ii.     \(\text{Scale factor (B to A)} = \dfrac{\text{Diameter A}}{\text{Diameter B}} = \dfrac{2.4}{0.8} = 3 \)

\(\text{Scale factor (Area)} = 3^2 = 9 \)

\(\therefore\ \text{Area of Circle A = 9 × Area of Circle B} \)

Similarity, SMB-025

A triangular prism is pictured below.
 

By what factor will its volume change if

  1. Each dimension is doubled?   (1 mark)

  2. Each dimension is decreased by two-thirds?   (2 marks)
Show Answers Only
  1. \(\text{Increases by a factor of 8}\)
  2. \(\text{Decreases by a factor of}\ \ \dfrac{1}{27} \)
Show Worked Solution

i.    \(\text{Dimensions increase by a factor of 2}\)

\(\Rightarrow\ \text{Volume increases by a factor of}\ 2^3 = 8\)
 

ii.    \(\text{Dimensions decrease by two-thirds}\)

\(\Rightarrow\ \text{i.e. adjust dimensions by a factor of}\ \ \dfrac{1}{3} \)

\(\Rightarrow\ \text{Volume decreases by a factor of}\ \Big{(} \dfrac{1}{3} \Big{)}^3 = \dfrac{1}{27} \)

Similarity, SMB-016

Triangle I and Triangle II are similar. Pairs of equal angles are shown.
 

Find the area of Triangle II?  (3 marks)

Show Answers Only

`24\ text{cm}^2`

Show Worked Solution

`text(In Triangle I, using Pythagoras:)`

`text{Base}` `= sqrt(5^2-3^2)`
  `= 4`

 
`text(Triangle I ||| Triangle II (given))`

♦♦ Mean mark 29%.

`=>\ text(corresponding sides are in the same ratio)`

`text{Scale factor}\ = 6/2=2`

`text{Scale factor (Area)}\ = 2^2=4`

`:. text(Area (Triangle II))` `= 4 xx text{Area of triangle I}`
  `= 4 xx 1/2 xx 3 xx 4`
  `=24\ text{cm}^2`

Similarity, SMB-012

Poppy uses a photocopier to enlarge this picture.
 

   

The enlarged picture is 3 times as high and 3 times as wide as the original.

By what factor is the area of the picture increased?   (2 marks)

Show Answers Only

`text(9 times the area of the original)`

Show Worked Solution

`text{Method 1}`

`text{Dimensions increased by a factor of 3}`

`:.\ text{Area increased by a factor of}\ 3^2 = 9`
 

`text{Method 2}`

`text(Area of original picture)\ = 3 xx 5 = 15\ text(cm)^2`

`text(Area of enlarged picture)\ = 9 xx 15 = 135\ text(cm)^2`

`:.\ text(Factor)\ = 135/15 = 9\ text(times)`

Volume, SMB-017

A deep ocean submarine is constructed in the shape of a sphere. 
 

If the volume of the sphere is 12.1 cubic metres, calculate its diameter in metres, correct to two decimal places.   (2 marks)

Show Answers Only

`2.85\ text(m)`

Show Worked Solution
`text{Volume}` `=4/3 xx pi xx r^3`  
`12.1` `=4/3 xx pi xx r^3`  
`r^3` `= \frac{3 xx 12.1}{4 xx pi}`  
  `=2.888`  
`r` `=1.424…\ text{m}`  

 
`:. text{Diameter}\ = 2 xx 1.242… = 2.85\ text{m (2 d.p.)}`

Volume, SMB-015

A funnel is made in the shape of a square cone with radius 9.5 centimetres and height 19.5 centimetres.
 

Find the volume of the funnel in cubic centimetres, giving your answer correct to 2 decimal places.   (2 marks)

Show Answers Only

`1842.94\ text(cm)^3`

Show Worked Solution
`text{Volume}` `= 1/3 xx A xx h`  
  `=1/3 xx pi xx 9.5^2 xx 19.5`  
  `= 1842.936…`  
  `=1842.94\ text{cm}^3`  

Volume, SMB-014

The storage building below is constructed by joining a square pyramid to a cube, with all measurements in metres.
 

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

Show Answers Only

`150\ text(m)^3`

Show Worked Solution
`text{Volume (cube)}` `= 5 xx 5 xx 5`  
  `=125\ text{m}^3`  

 

`text{Volume (pyramid)}` `= 1/3 A h`
  `=  1/3 xx 5 xx 5 xx 3`
  `= 25\ text(cm)^3`

 
`text{Total volume}\ = 125 + 25 = 150\ text{m}^3`

Volume, SMB-013

The square pyramid below, has a side measurement of 120 metres and a perpendicular height `(h)` of 65 metres.
 

Find the volume of the pyramid in cubic metres.   (2 marks)

Show Answers Only

`312\ 000\ text(m)^3`

Show Worked Solution
`text{Volume}` `= 1/3 xx A xx h`  
  `=1/3 xx 120 xx 120 xx 65`  
  `=312\ 000\ text{m}^3`  

Volume, SMB-007

A cannon ball is made out of steel and has a diameter of 23 cm.

Find the volume of the sphere in cubic centimetres (correct to 1 decimal place).  (2 marks)

Show Answers Only

`6370.6\ text{cm}^3`

Show Worked Solution

`text(Radius)= 23/2 = 11.5\ text(cm)`

`text(Volume)` `= 4/3pir^3`
  `= 4/3 xx pi xx 11.5^3`
  `= 6370.626…`
  `= 6370.6\ text{cm}^3\ text{(to 1 d.p.)}`

Volume, SMB-006

A concrete water pipe is manufactured in the shape of an annular cylinder. The dimensions are shown in the diagrams.
 


 

Find the approximate volume of concrete needed to make the water pipe, giving your answer in cubic metres correct to two decimal places.   (3 marks)

Show Answers Only

`0.70\ text(m)^3`

Show Worked Solution
`text(Volume)` `= text(Area of annulus) xx h`
  `= (piR^2 – pir^2) xx 2.8`
  `= (pi xx 0.45^2 – pi xx 0.35^2) xx 2.8`
  `= 0.7037…`
  `= 0.70\ text(m)^3`

Volume, SMB-005

Two identical spheres fit exactly inside a cylindrical container, as shown.
 

The diameter of each sphere is 12 cm.

 What is the volume of the cylindrical container, to the nearest cubic centimetre?   (3 marks)

Show Answers Only

`2714\ text{cm³}`

Show Worked Solution

`text(S)text(ince diameter sphere = 12 cm) `

`=>\ text(Radius of cylinder = 6 cm)`

`text(Height of cylinder)` `= 2 xx text(diameter of sphere)`
  `= 2 xx 12`
  `= 24\ text(cm)`
   
`:.\ text(Volume cylinder)` `= pi r^2 h`
  `= pi xx 6^2 xx 24`
  `= 2714.336…`
  `= 2714\ text{cm³}`

Area, SMB-036

Find the surface area of the solid pictured below which is composed of a right cone with a hemisphere attached to the base. Give your answer to the nearest square centimetre.  (3 marks)
  

Show Answers Only

\(\ 616\ \text{cm}^2\)

Show Worked Solution
\(\text{SA (hemisphere)}\) \(= \dfrac{1}{2} \times 4 \pi r^2\)  
  \(= \dfrac{1}{2} \times 4 \pi \times 7^2\)  
  \(=307.87…\ \text{cm}^2\)  

 

\(\text{SA (cone)}\) \(= \pi rl\)  
  \(= \pi \times 7 \times 14\)  
  \(=307.87…\ \text{cm}^2\)  

 
\(\text{Total SA}\ = 2 \times 307.87… = 616\ \text{cm}^2\ \ (\text{nearest cm}^2) \)

Area, SMB-018

A right cone with perpendicular height of 8 cm, slant height of 10 cm and a base diameter of 12 cm is pictured below.
 

Find the total surface area of the cone, including its base, correct to two decimal places.   (3 marks)

Show Answers Only

\(301.59\ \text{cm}^2 \)

Show Worked Solution

\(r = 6\ \text{cm}, \ l =  10\ \text{cm} \)

\(\text{SA}\) \(= \pi r^2 + \pi r l\)  
  \(=  \pi \times 6^2 + \pi \times 6 \times 10 \)  
  \(=301.592… \)  
  \(=301.59\ \text{cm}^2\ \text{(2 d.p.)}\)  

Area, SMB-017

A square pyramid with a slant height of 15 centimetres is pictured below.
 

Find the total surface area of the pyramid, including its base.   (2 marks)

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\(400\ \text{cm}^2 \)

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\(\text{SA}\) \(= (10 \times 10) + 4 \times (\dfrac{1}{2} \times 10 \times 15)\)  
  \(=100 + 4(75) \)  
  \(=400\ \text{cm}^2 \)  

Area, SMB-016

A square pyramid with a perpendicular height of 8 metres is pictured below.
 

  1. Using Pythagoras, calculate the slant height of the pyramid.   (2 marks)
  1. Find the total surface area of the pyramid, including its base, to the nearest square metre.   (2 marks)
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i.    \(10\ \text{m} \)

ii.    \(384\ \text{m}^2 \)

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i.    \(\text{Let}\ \ s =\ \text{slant height}\)

\(s^2\) \(=6^2 + 8^2\)  
  \(=100\)  
\(s\) \(=10\ \text{m}\)  

 

ii.    \(\text{SA}\) \(= (12 \times 12) + 4 \times (\dfrac{1}{2} \times 12 \times 10)\)  
  \(=144 + 4(60) \)  
  \(=384\ \text{m}^2 \)  

Area, SMB-030

A glass aviary is made up of four triangles and a square, as shown in the diagram below.

Harry is hired to clean the interior sides of the aviary, not including the floor.

What is the area that Harry will need to clean?   (2 marks)

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`402\ text(m)^2`

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`text(Area to clean)`

`= 4 xx 1/2 bh`

`= 4 xx 1/2 xx 15 xx 13.4`

`= 402\ text(m)^2`