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Volume, SM-Bank 136

  1. What shape best describes the uniform cross-section of this prism?  (1 mark)

  2. Calculate the volume of the prism in cubic centimetres.  (2 marks)
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a.    \(\text{A rhombus}\)

b.    \(497.664\ \text{cm}^3\)

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a.    \(\text{The uniform cross-section is a rhombus.}\)

b.    \(\text{Area of rhombus cross-section }\)

\(A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times 9.6\times 7.2\)
  \(=34.56\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=34.56\times 14.4\)
  \(=497.664\ \text{cm}^3\)

Volume, SM-Bank 135

The figure below is a prism with a rhombus as the uniform cross-section.

Calculate the value of \(x\), the length of the diagonal in the cross-section, given the volume of the prism is 1950 cubic metres.  (2 marks)

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\(20\ \text{m}\)

Show Worked Solution

\(\text{The uniform cross-section is a rhombus.}\)

\(\therefore\ A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times x\times 15\)
  \(=7.5x\ \text{m}^2\)

 

\(V\) \(=A\times h\)
\(1950\) \(=7.5x\times 13\)
\(1950\) \(=97.5x\)
\(\therefore\ x\) \(=\dfrac{1950}{97.5}\)
  \(=20\ \text{m}\)

Volume, SM-Bank 134

The figure below is a prism with a kite as the uniform cross-section.

Calculate the value of \(x\), the length of the diagonal in the cross-section, given the volume of the prism is 1485 cubic centimetres.  (2 marks)

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\(11\ \text{cm}\)

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\(\text{The uniform cross-section is a kite.}\)

\(\therefore\ A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times x\times 18\)
  \(=9x\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
\(1485\) \(=9x\times 15\)
\(1485\) \(=135x\)
\(\therefore\ x\) \(=\dfrac{1485}{135}\)
  \(=11\ \text{cm}\)

Volume, SM-Bank 133

The figure below is a prism with a parallelogram as the uniform cross-section.

Calculate the value of \(x\), the perpendicular height of the cross-section, given the volume of the prism is 2880 cubic millimetres.  (2 marks)

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\(12\ \text{mm}\)

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\(\text{The uniform cross-section is a parallelogram.}\)

\(\therefore\ A\) \(=b\times h\)
  \(=10\times x\)
  \(=10x\ \text{mm}^2\)

 

\(V\) \(=A\times h\)
\(2880\) \(=10x\times 24\)
\(2880\) \(=240x\)
\(\therefore\ x\) \(=\dfrac{2880}{240}\)
  \(=12\ \text{mm}\)

Volume, SM-Bank 132

  1. What shape best describes the uniform cross-section of this prism?  (1 mark)

  2. Calculate the volume of the prism in cubic centimetres.  (2 marks)
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a.    \(\text{A parallelogram}\)

b.    \(279.972\ \text{cm}^3\)

Show Worked Solution

a.    \(\text{The uniform cross-section is a parallelogram.}\)

b.    \(\text{Area of parallelogram cross-section }\)

\(A\) \(=b\times h\)
  \(=6.6\times 4.2\)
  \(=27.72\ \text{cm}^2\)

 

\(V\) \(=A\times h\)
  \(=27.72\times 10.1\)
  \(=279.972\ \text{cm}^3\)

Volume, SM-Bank 131

  1. What shape best describes the uniform cross-section of this prism?  (1 mark)

  2. Calculate the volume of the prism in cubic metres.  (2 marks)
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a.    \(\text{A kite}\)

b.    \(28.8\ \text{m}^3\)

Show Worked Solution

a.    \(\text{The uniform cross-section is a kite.}\)

b.    \(\text{Area of kite cross-section }\)

\(A\) \(=\dfrac{1}{2}\times x\times y\)
  \(=\dfrac{1}{2}\times 3\times 4.8\)
  \(=7.2\ \text{m}^2\)

 

\(V\) \(=A\times h\)
  \(=7.2\times 4\)
  \(=28.8\ \text{m}^3\)

Volume, SM-Bank 123

Calculate the volume of the prism below in cubic metres.  (2 marks)

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\(540\ \text{m}^3\)

Show Worked Solution
\(\text{Area of cross-section}\) \(=(9\times 8)-(3\times 6)\)
  \(=72-18\)
  \(=54\ \text{m}^2\)

 

\(V\) \(=A\times h\)
  \(=54\times 10\)
  \(=540\ \text{m}^3\)

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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\(258\ \text{bricks}\)

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

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

\(=294\)

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

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

\(=36\)

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

Volume, SM-Bank 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.    \(100\ \text{cm}^3\)

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

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)

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

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\(\text{Area of trapezium}\)

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

 

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

Volume, SM-Bank 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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\(80\ \text{cm}^2\)

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

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

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

Volume, SM-Bank 011

The floor of a chicken coop is in the shape of a trapezium.

The floor, \(ABCD\), and the chicken coop are shown below.
 

 

In the diagram \(AB = 3\ \text{m}, BC = 2\ \text{m and}\ \ CD = 5\ \text{m}.\)
 

  1. What is the area of the floor of the chicken coop?

     

    Write your answer in square metres.  (2 marks)


  2. If the height of the chicken coop is 2.4 metres, calculate the volume in cubic metres.  (1 mark)

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a.    \(8\ \text{m}^2\)

b.    \(19.2\ \text{m}^3\)

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a.     
\(A\) \(=\dfrac{h}{2}(a+b)\)
    \(=\dfrac{2}{2}(3+5)\)
    \(= 8\ \text{m}^2\)

 

b.     \(V\) \(=Ah\)
    \(=8\times 2.4\)
    \(=19.2\ \text{m}^3\)

Volume, SM-Bank 009

A shed has the shape of a prism. Its front face, \(AOBCD\), is shaded in the diagram below.

\(ABCD\) is a rectangle and \(M\) is the midpoint of \(AB\).

\(\Delta AMO\) is right angled and the length of \(AM\) is 3 metres.
 

GEOMETRY, FUR2 2008 VCAA 2

  1. Using Pythagoras' Theorem find the length of \(OM\).  (2 marks)


  2. Calculate the area of the front face of the shed, \(AOBCD\), in square metres.  (2 marks)


  3. Find the volume of the shed in cubic metres.  (1 mark)


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a.    \(1.6\ \text{m}\)

b.    \(18\ \text{m}^2\)

c.    \(180\ \text{m}^3\)

Show Worked Solution
a.  

\(\text{In}\ \Delta AOM,\ \text{using Pythagoras:}\)

\(OM^2+AM^2\) \(=OA^2\)
\(OM^2+3^2\) \(=3.4^2\)
\(OM^2\) \(=3.4^2-3^2\)
\(OM\) \(=\sqrt{3.4^2-3^2}\)
  \(=\sqrt{2.56}\)
  \(=1.6\ \text{m}\)

 

b.   \(\text{Area of front face of shed}\)

\(=\text{Area}\ \Delta AOB+\text{Area}\ ABCD\)

\(=\dfrac{1}{2}\times 1.6\times 6+2.2\times 6\)

\(=18\ \text{m}^2\)

 

c.    \(V\) \(=Ah\)
    \(=18\times 10\)
    \(=180\ \text{m}^3\)

Volume, SM-Bank 001 MC

The building shown in the diagram is 8 m wide and 24 m long.

The side walls are 4 m high.

The peak of the roof is 6 m vertically above the ground.

In cubic metres, the volume of this building is

  1. \(384\)
  2. \(576\)
  3. \(960\)
  4. \(1152\)
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\(C\)

Show Worked Solution

 

\(\text{Area of front of house}\)

\(=(4\times 8)+\dfrac{1}{2} (8\times 2)\)

\(=32+8\)

\(=40\ \text{m}^2\)

\(V\) \(=Ah\)
  \(=40\times 24\)
  \(=960\ \text{m}^3\)

 
\(\Rightarrow C\)