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Volume, SM-Bank 113

Convert 9.8 kilolitres to litres.  (1 mark)

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\(9800\ \text{L}\)

Show Worked Solution
\(1\ \text{kL}\) \(=1000\ \text{L}\)
\(\therefore\ 9.8\ \text{kL}\) \(=9.8\times 1000\ \text{L}\)
  \(=9800\ \text{L}\)

Volume, SM-Bank 112 MC

A factory worker pours 800 millilitre bottles of barbecue sauce into a container that can hold 9.6 litres in total.

Which one of these expressions shows how many bottles of barbecue sauce will be needed to fill the container?

  1. \(9.6\times 800\)
  2. \(9600\times 800\)
  3. \(9600\ ÷\ 800\)
  4. \(800\ ÷\ 9.6\)

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\(C\)

Show Worked Solution

\(\text{9.6 L = 9600 millilitres}\)

\(\text{Bottles of barbecue sauce}=9600\ ÷\ 800\)
   
\(\Rightarrow C\)

Volume, SM-Bank 111

A water cooler has a capacity of 8.55 L.

How many millilitres does the water cooler hold when it is full?  (1 mark)

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\(8550\ \text{mL}\)

Show Worked Solution

\(\text{Coverting litres to mL:}\)

\(8.55\ \text{L}\times 1000=8550\ \text{mL}\)

Volume, SM-Bank 110

A container has some water in it.
 

 
An extra 300 mL of water is added to the container.

How many millilitres (mL) of water will then be in the container?  (2 marks)

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\(\text{1550 mL}\)

Show Worked Solution

\(\text{Note: Each division in the jug is 250 mL}\)

\(\text{Water in jug}\) \(=1250+300\)
  \(=1550\ \text{mL}\)

Volume, SM-Bank 109

A water container has 5 litres of water in it.

Kate pours water into her dog's bowl.

She pours the water into the 250 cubic centimetre bowl until it is full.
 


 

How much water is left in the container?  (2 marks)

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\(\text{4750 millilitres}\)

Show Worked Solution

\(\text{1 cm}^3 = 1\ \text{mL}\Longrightarrow \text{250 cm}^3 = 250\ \text{mL} \)

\(\text{5 litres = 5000 mL}\)

\(\therefore\ \text{Water left}\) \(=5000-250\)
  \(=4750\ \text{millilitres}\)

Volume, SM-Bank 108

A class is making ice-cubes for a science experiment.

One ice-cube container requires 0.35 litres of water to fill it.

How many millilitres of water would a student need to fill up one container?  (1 mark)

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\(350\)

Show Worked Solution

\(\text{Since there are 1000 mL in 1 litre,}\)

\(\text{Volume}\) \(=0.35\times 1000\)
  \(= 350\ \text{millilitres}\)

Volume, SM-Bank 107 MC

A glass of water containing 255 mL of water is poured into a jug that already contains 1.65 L of water in it.

How much water is now in the jug?

  1. 256.65 mL
  2. 420 mL
  3. 1.805 mL
  4. 1.905 mL
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\(D\)

Show Worked Solution
\(1.65\ \text{L} + 255\ \text{mL}\) \(=1.65\ \text{L} + 0.255\ \text{L}\)
  \(=1.905\ \text{L}\)

 
\(\Rightarrow D\)

Volume, SM-Bank 106 MC

A petrol container has a capacity of 10.25 L.

How many millilitres does the petrol container hold when it is full?

  1. 1025
  2. 10 025
  3. 10 250
  4. 102 500
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\(C\)

Show Worked Solution

\(\text{1 litre = 1000 mL}\)

\(\therefore 10.25\ \text{L}\times 1000 = 10\ 250\ \text{mL}\)

\(\Rightarrow C\)

Volume, SM-Bank 105 MC

Karen is filling her pool with water.

Which unit would be the most appropriate to measure the volume of water she needs to fill the pool?

  1. kilograms
  2. kilojoules
  3. kilolitres
  4. kilometres
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\(C\)

Show Worked Solution

\(\text{The unit must measure liquid }\Longrightarrow \text{ kilolitres}\)

\(\Rightarrow C\)

Volume, SM-Bank 104 MC

Fiona is a nurse who is administering vaccines to patients using a needle.

Which unit would be the most appropriate to measure the volume of vaccine she needs to inject?

  1. milligrams
  2. joules
  3. millimetres
  4. millilitres
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\(D\)

Show Worked Solution

\(\text{A needle injects liquid vaccine into people.}\)

\(\text{The most appropriate measure: millilitres}\)

\(\Rightarrow D\)

Volume, SM-Bank 103 MC

Carrie has a small container of milk.

It contains 250 millilitres of milk.
 

Carrie buys a pack of 6 of these milk containers.

How many litres of milk are in the pack?

  1. 0.25 litres
  2. 1.5 litres
  3. 2.5 litres
  4. 15 litres
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\(B\)

Show Worked Solution

\(\text{Note: 1 L = 1000 mL}\)

\(\text{Total volume in pack}\) \(=6\times 250\)
  \(=1500\ \text{mL}\)
  \(=1.5\ \text{litres}\)

 
\(\Rightarrow B\)

Volume, SM-Bank 096

Guy builds a brick structure that is pictured below.

The structure is 7 bricks high, 7 bricks wide and 6 bricks deep.

The structure is solid brick but has a hole that goes from one side to the other which is 3 bricks high and two bricks wide, as shown in the diagram.
 

 
How many bricks are in the stack?  (2 marks)

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\(258\ \text{bricks}\)

Show Worked Solution

\(\text{Bricks in the stack if no hole}\)

\(=7\times 7\times 6\)

\(=294\)

\(\text{Bricks removed to make hole}\)

\(=3\times 2\times 6\)

\(=36\)

\(\therefore\ \text{Bricks in stack}\) \(=294-36\)
  \(=258\)

Volume, SM-Bank 095

A horse trough is in the shape of a rectangular prism, pictured below.
 

  1.  What is the volume of the prism in cubic centimetres?  (2 marks)
  2. What is the capacity of the horse trough in litres?  (1 mark)
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a.    \(160\ 000\ \text{cm}^3\)

b.    \(160\ \text{L}\)

Show Worked Solution
a.    \(\text{Volume}\) \(=Ah\)
    \(=(40\times 50)\times 80\)
    \(=160\ 000\ \text{cm}^3\)

  
b.    \(1000\ \text{cm}^3=1\ \text{Litre}\)

\(\therefore\ \text{Capacity}\) \(=\dfrac{160\ 000}{1000}\)
  \(=160\ \text{L}\)

Volume, SM-Bank 094

Two identical solid cubes are placed at the bottom of a fish tank.
 

 

The fish tank is then completely filled, as shown below.

What is the volume of the water that surrounds the cubes?

Give your answer in cubic centimetres.  (2 marks)

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\(185\ 250\ \text{cm}^3\)

Show Worked Solution
\(\text{Volume of tank}\) \(=l\times b\times h\)
  \(=80\times 60\times 40\)
  \(=192\ 000\ \text{cm}^3\)

 

\(\text{Volume of cubes}\) \(=2\times s^3\)
  \(=2\times 15^3\)
  \(=6750\ \text{cm}^3\)

 

\(\therefore\ \text{Volume of water}\) \(=192\ 000-6750\)
  \(=185\ 250\ \text{cm}^3\)

Volume, SM-Bank 093

Two views of a trapezoidal prism are shown below.
 

Each square on this grid has an area of one square centimetre.

The vertical edges of the prism are 5 centimetres.

  1. What is the area of the shaded cross-section in square centimetres?  (2 marks)
  2. What is the volume of the prism in cubic centimetres?  (2 marks)
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a.    \(20\ \text{cm}^2\)

b.    \(100\ \text{cm}^3\)

Show Worked Solution

a.    \(\text{The area of the cross-section (trapezium)}\)

\(=14\ \text{squares}+8\ \text{triangles}\ (1\times 1) + 1\ \text{triangle}\ (1\times 4)\)

\(=14+(8\times \dfrac{1}{2})+2\)

\(=20\ \text{cm}^2\)
 

b.    \(\text{Volume}\) \(=Ah\)
    \(=20\times 5\)
    \(=100\ \text{cm}^3\)

Volume, SM-Bank 092

Two views of a trapezoidal prism are shown below.

Each square on this grid has an area of one square centimetre.

The vertical edges of the prism are 4 centimetres.

  1. What is the area of the shaded cross-section in square centimetres?  (2 marks)
  2. What is the volume of the prism in cubic centimetres?  (2 marks)
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a.    \(18\ \text{cm}^2\)

b.    \(72\ \text{cm}^3\)

Show Worked Solution

a.    \(\text{The area of the cross-section (trapezium)}\)

\(=12\ \text{squares}+9\ \text{triangles}\ (1\times 1) + 1\ \text{triangle}\ (1\times 3)\)

\(=12+(9\times \dfrac{1}{2})+1\dfrac{1}{2}\)

\(=18\ \text{cm}^2\)
 

b.    \(\text{Volume}\) \(=Ah\)
    \(=18\times 4\)
    \(=72\ \text{cm}^3\)

Properties of Geometric Figures, SM-Bank 026

Determine the value of \(a^{\circ}\), \(b^{\circ}\), and \(c^{\circ}\), giving reasons for your answer.   (3 marks)

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\(\text{All radii are equal (see diagram).}\)

\(a^{\circ} = 70^{\circ}\ \ \text{(angles opposite equal sides in isosceles triangle)} \)

\(b^{\circ} = 2 \times 70 = 140^{\circ}\ \ \text{(external angle = sum of interior opposite angles)} \)

\(140^{\circ} + 2 \times c^{\circ}\) \(=180^{\circ}\ \ \text{(angle sum of isosceles triangle)} \)  
\(2c^{\circ}\) \(=180-40\)  
\(c^{\circ}\) \(=\dfrac{40}{2} = 20^{\circ} \)  
Show Worked Solution

\(\text{All radii are equal (see diagram).}\)

\(a^{\circ} = 70^{\circ}\ \ \text{(angles opposite equal sides in isosceles triangle)} \)

\(b^{\circ} = 2 \times 70 = 140^{\circ}\ \ \text{(external angle = sum of interior opposite angles)} \)

\(140^{\circ} + 2 \times c^{\circ}\) \(=180^{\circ}\ \ \text{(angle sum of isosceles triangle)} \)  
\(2c^{\circ}\) \(=180-40\)  
\(c^{\circ}\) \(=\dfrac{40}{2} = 20^{\circ} \)  

Properties of Geometric Figures, SM-Bank 025

  1. Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

  2. Determine the value of \(b^{\circ}\), giving reasons for your answer.   (2 marks)
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a.    


 

\(\text{All radii are equal (see diagram).}\)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(a^{\circ} = 180-(2 \times 60)=60^{\circ}\ \ \text{(angle sum of triangle)} \)
 

b.   \(c^{\circ} = 180-60=120^{\circ}\ \ \text{(180° in straight line)} \)

\(120^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  
Show Worked Solution

a.    


 

\(\text{All radii are equal (see diagram).}\)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(a^{\circ} = 180-(2 \times 60)=60^{\circ}\ \ \text{(angle sum of triangle)} \)
 

b.   \(c^{\circ} = 180-60=120^{\circ}\ \ \text{(180° in straight line)} \)

\(120^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  

Properties of Geometric Figures, SM-Bank 024

An isosceles triangle is pictured below.
 

Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

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\(b^{\circ} = 180-95=85^{\circ}\ \ \text{(180° in straight line)} \)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  
Show Worked Solution

\(b^{\circ} = 180-95=85^{\circ}\ \ \text{(180° in straight line)} \)

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=85\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= \dfrac{85}{2}\)  
  \(=42.5^{\circ}\)  

Properties of Geometric Figures, SM-Bank 023

Find the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

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\(a^{\circ}+67^{\circ}\) \(=108\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= 108-67\)  
  \(=41^{\circ}\)  
Show Worked Solution
\(a^{\circ}+67^{\circ}\) \(=108\ \ \text{(external angle = sum of interior opposite angles)} \)  
\(a^{\circ}\) \(= 108-67\)  
  \(=41^{\circ}\)  

Properties of Geometric Figures, SM-Bank 022

The diagram below shows an isosceles triangle.
 

Find the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(2y^{\circ}\) \(=180-32\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{148}{2}\)  
  \(=74^{\circ}\)  
\(x^{\circ}\) \(=32+74\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=106^{\circ}\)  
Show Worked Solution

\(2y^{\circ}\) \(=180-32\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{148}{2}\)  
  \(=74^{\circ}\)  
\(x^{\circ}\) \(=32+74\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=106^{\circ}\)  

Properties of Geometric Figures, SM-Bank 021

 

Find the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(2y^{\circ}\) \(=180-78\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{102}{2}\)  
  \(=51^{\circ}\)  
\(x^{\circ}\) \(=78+51\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=129^{\circ}\)  
Show Worked Solution

\(2y^{\circ}\) \(=180-78\ \ \text{(angles opposite equal sides in isosceles triangle)} \)  
\(y^{\circ}\) \(=\dfrac{102}{2}\)  
  \(=51^{\circ}\)  
\(x^{\circ}\) \(=78+51\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=129^{\circ}\)  

Properties of Geometric Figures, SM-Bank 011

In the diagram, \(AB\) is parallel to \(DE\).
 

  1. On the diagram, label the alternate angles to \(a^{\circ}\) and \(b^{\circ}\).   (1 mark)

  2. Using part (a), show that the sum of internal angles of a triangle equals 180°.   (2 marks)
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a.    
     

b.    \(DE\ \text{is a straight line.}\)

\(a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\ \ \text{(180° in a straight line)}\)

\(\therefore \ \text{Angle sum of}\ \Delta = a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\)

Show Worked Solution

a.    
       

b.    \(DE\ \text{is a straight line.}\)

\(a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\ \ \text{(180° in a straight line)}\)

\(\therefore \ \text{Angle sum of}\ \Delta = a^{\circ} + b^{\circ} + c^{\circ} = 180^{\circ}\)

Volume, SM-Bank 091

An ancient building has the shape of a trapezoidal prism.

The shaded side is a trapezium.
 

 
What is the volume of the building in m³ ?  (2 marks)

Show Answers Only

\(1344\ \text{m}^3\)

Show Worked Solution

\(\text{Area of trapezium}\)

\(A\) \(=\dfrac{h}{2}\times (a+b)\)
  \(=\dfrac{6}{2}\times (12+16)\)
  \(=3\times 28\)
  \(=84\ \text{m}^2\)

 

\(\therefore\ V\) \(=Ah\)
  \(=84\times 16\)
  \(=1344\ \text{m}^3\)

Volume, SM-Bank 090

A rectangular trough in a paddock provides water for horses.

Its measurements can be seen below:
  

  1. Calculate the volume of the trough in cubic metres.  (2 marks)
  2. Given that one cubic metre holds 1000 litres of water, what is the capacity of the trough in litres?  (1 mark)

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a.    \(3\ \text{m}^3\)

b.    \(\text{3000 litres}\)

Show Worked Solution
a.    \(\text{Volume}\) \(=Ah\)
    \(=(0.5\times 0.75)\times 8\)
    \(=0.375\times 8\)
    \(=3\ \text{m}^3\)

 

b.    \(\text{Capacity}\) \(=1000\times 3\)
    \(=3000\ \text{litres}\)

Properties of Geometrical Figures, SM-Bank 018

The diagram below shows a right-angled triangle.
 

Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

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\(138^{\circ}\)

Show Worked Solution

\(\text{Right angle}\ = 90^{\circ} \)

\(a^{\circ}\) \(=48+90\ \ \text{(external angle = sum of interior opposite angles)} \)  
  \(=138^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 017

The diagram below shows an isosceles triangle.
 

Determine the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(40^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(x^{\circ}\) \(=180-(2 \times 70)\ \ \text{(180° in triangle)} \)  
  \(=180-140\)  
  \(=40^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 016

The diagram below shows an isosceles triangle.
 

Determine the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(110^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(x^{\circ}\) \(=180-(2 \times 35)\ \ \text{(180° in triangle)} \)  
  \(=110^{\circ}\)  

Volume, SM-Bank 089

A large sculpture is made in the shape of a cube.

The total length of all of its edges is 60 metres.

What is the volume of the cube in cubic metres?  (2 marks)

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\(125\ \text{m}^3\)

Show Worked Solution

\(\text{A cube has 12 edges.}\)

\(\text{Length of 1 edge} =\dfrac{60}{12} = 5\ \text{m}\)
 

\(\therefore\ \text{Volume of cube}\) \(=5\times 5\times 5\)
  \(=125\ \text{m}^3\)

Properties of Geometrical Figures, SM-Bank 015

The diagram below shows an isosceles triangle.
 

Determine the value of \(a^{\circ}\), giving reasons for your answer.   (2 marks)

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\(71^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2a^{\circ}\) \(=180-38\ \ \text{(180° in triangle)} \)  
\(a^{\circ}\) \(=\dfrac{142}{2}\)  
  \(=71^{\circ}\)  

Properties of Geometrical Figures, SM-Bank 014

The diagram below shows an isosceles triangle.
 

Determine the value of \(x^{\circ}\), giving reasons for your answer.   (2 marks)

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\(59^{\circ}\)

Show Worked Solution

\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal}\)

\(2x^{\circ}\) \(=180-62\ \ \text{(180° in triangle)} \)  
\(x^{\circ}\) \(=\dfrac{118}{2}\)  
  \(=59^{\circ}\)  

Volume, SM Bank 088 MC

Alan is moving house and is packing his belongings in rectangular cardboard boxes.

The height of each box is 0.5 metres.

Which box has a volume of 0.12 cubic metres?
 

 

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\(D\)

Show Worked Solution

\(\text{Volume}\)

\(=0.4\times 0.6\times 0.5\)

\(= 0.12\ \text{m}^3\)

\(\Rightarrow D\)

Volume, SM-Bank 087 MC

Two bricks can be joined to make three different rectangular prisms. Two of them are shown here.
 

 
What would be the measurements of the third prism?

  1. 18 cm by 16 cm by 7 cm
  2. 36 cm by 8 cm by 7 cm
  3. 32 cm by 18 cm by 7 cm
  4. 36 cm by 14 cm by 8 cm
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\(B\)

Show Worked Solution

\(\text{36 cm by 8 cm by 7 cm}\)
 

\(\Rightarrow B\)

Volume, SM-Bank 080

Wes made a small model staircase by stacking blocks.

There are no gaps between blocks.
 

 If each block is 1 cubic centimetre, what is the volume of the model staircase?  (2 marks)

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\(80\ \text{cm}^2\)

Show Worked Solution

\(\text{Area of cross-section = 20 cm}^2\)

\(V\) \(=Ah\)
  \(=20\times 4\)
  \(=80\ \text{cm}^3\)

 
\(\therefore\ \text{The volume of the model staircase = 80 cm}^3\)

Volume, SM-Bank 079 MC

A sculpture is pictured from 3 different angles below:
 

 
The sculpture is in the shape of

  1. a trapezium and a circle.
  2. a trapezium and a cylinder.
  3. a sphere and a prism.
  4. a cylinder and a prism.
Show Answers Only

\(D\)

Show Worked Solution

\(\text{The bottom shape is a trapezoidal prism (not a trapezium).}\)

\(\therefore\ \text{The sculpture is in the shape of a cylinder and a prism.}\)

\(\Rightarrow D\)