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Measurement, STD2 EQ-Bank 14 MC

A square pyramid has a volume of 384 cm³ and a perpendicular height of 8 cm.

 Calculate the side length of the square base. Use the formula \(V = \frac{1}{3}Ah\) where \(A\) is the area of the base.

  1. 8 cm
  2. 12 cm
  3. 16 cm
  4. 24 cm
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\(B\)

Show Worked Solution
\(V\) \(=\frac{1}{3}Ah\)
\(384\) \(=\frac{1}{3}\times A\times 8\)
\(384\) \(=\dfrac{8A}{3}\)
\(8A\) \(=384\times 3=1152\)
\(A\) \(=\dfrac{1152}{8}\)
  \(=144\ \text{cm}^2\)
\(\text{Since the base is a square: }x^2\) \(=144\)
\(x\) \(=\sqrt{144}\)
  \(=12\ \text{cm}\)

 

\(\Rightarrow B\)

Measurement, STD2 EQ-Bank 13 MC

A square pyramid has a base with side length 9 cm and a perpendicular height of 8 cm.

Calculate the volume of the pyramid. 

  1. 72 cm³
  2. 144 cm³
  3. 192 cm³
  4. 216 cm³
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\(D\)

Show Worked Solution

\(A=9^2=81\)

\(V\) \(=\frac{1}{3}Ah\)
  \(=\frac{1}{3}\times 81\times 8\)
  \(=216\ \text{cm}^3\)

  
\(\Rightarrow D\)

Measurement, STD2 EQ-Bank 7 MC

A conical water tank has a radius of 3 metres and a perpendicular height of 5 metres.

Calculate the volume of water the tank can hold in kilolitres. (Note: 1 m³ = 1 kL)

  1. 15.71 kL
  2. 23.56 kL
  3. 47.12 kL
  4. 141.37 kL
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\(C\)

Show Worked Solution
\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\times \pi \times (3)^2\times 5\)
  \(=47.123\dots\)

 
\(\text{Using: 1 m}^3 = 1\ \text{kL:} \)

\(\text{47.12 m}^3= 47.12\ \text{kL}\)

\(\Rightarrow C\)

Measurement, STD2 EQ-Bank 4 MC

A cone has a radius of 6 cm and a height of 8 cm.

Calculate the volume of the cone, giving your answer in cubic centimetres correct to 1 decimal place.

  1. 100.5 cm³
  2. 150.8 cm³
  3. 301.6 cm³
  4. 904.8 cm³
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\(C\)

Show Worked Solution
\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\times \pi \times (6)^2\times 8\)
  \(=301.592\dots\)
  \(=301.6\ \text{cm}^3 \text{(1 d.p.)}\)

  
\(\Rightarrow C\)

Measurement, STD2 EQ-Bank 19

The cone below has a diameter of 13.5 centimetres and a height of 0.4 metres.
 

Calculate the volume of the cone, giving your answer in cubic centimetres correct to 1 decimal place.   (2 marks)

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\(1908.5\ \text{cm}^3 \text{(1 d.p.)}\)

Show Worked Solution

\(\text{Radius}\ =\dfrac{13.5}{2}=6.75\ \text{cm}\)

\(0.4\ \text{m}\ =\ 40\ \text{cm}\)

\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\times \pi \times (6.75)^2\times 40\)
  \(=1908.517\dots\)
  \(=1908.5\ \text{cm}^3 \text{(1 d.p.)}\)

Measurement, STD2 EQ-Bank 18

The cone below has a radius of 4.2 metres and a height of 6.1 metres.
 

Calculate the volume of the cone, giving your answer in cubic metres correct to 1 decimal point.   (2 marks)

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\(112.7\ \text{m}^3 \text{(1 d.p.)}\)

Show Worked Solution
\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\times \pi \times (4.2)^2\times 6.1\)
  \(=112.682\dots\)
  \(=112.7\ \text{m}^3 \text{(1 d.p.)}\)

Measurement, STD2 EQ-Bank 16

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)

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\(312\, 000\ \text{m}^3\)

Show Worked Solution
\(V\) \(=\frac{1}{3}Ah\)
  \(=\frac{1}{3}\times 120\times 120\times 65\)
  \(=312\, 000\ \text{m}^3\)

Measurement, STD2 EQ-Bank 23

A cone with a radius 9.5 cm and height 19.5 cm is shown.
 

The cone is filled with liquid. 

Find the amount of liquid in the cone. Give your answer in litres correct to 3 decimal places.   (3 marks)

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\(1.843\ \text{L}\)

Show Worked Solution
\(V\) \(=\frac{1}{3}\pi r^2h\)
  \(=\frac{1}{3}\pi (9.5)^2\times 19.5\)
  \(=1842.93679\ \text{cm}^3\)

 
\(\text{Since 1 cm\(^3\) = 1 mL  and  1 L = 1000 mL:}\)

\(\text{Liquid in cone}=\dfrac{1842.9…}{1000}=1.8429…=1.843\ \text{litres (3 d.p.)}\)

Measurement, STD2 M1 2025 HSC 32

Solid spheres are placed inside a square-based pyramid as shown.
 

The base of the pyramid has side lengths of 14 cm . The height of the pyramid is \(h\) cm. The radius of each sphere is 1.5 cm.

The amount of empty space remaining inside the pyramid after 30 spheres have been placed inside the pyramid is 634 cm³.

What is the height, \(h\), of the pyramid? Give your answer correct to the nearest centimetre.   (3 marks)

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\(h=16 \ \text{cm}\)

Show Worked Solution

\(\text{Volume (pyramid)}=\dfrac{1}{3} \times A h=\dfrac{1}{3} \times 14 \times 14 \times h=\dfrac{196}{3} h\)

\(\text{Volume (spheres)}=30 \times \dfrac{4}{3} \pi\left(\dfrac{3}{2}\right)^3=135 \pi\)

\(\text{Empty space }=634 \ \text {(given)}\)

♦ Mean mark 47%.

\(\text{Equating the volumes:}\)

\(\dfrac{196}{3} h\) \(=135 \pi+634\)
\(h\) \(=(135 \pi+634) \times \dfrac{3}{196}\)
  \(=16.19 \ldots\)
  \(=16 \ \text{cm (nearest cm)}\)

Measurement, STD2 M1 2018 HSC 13 MC

A rectangular pyramid has base side lengths `3x` and `4x`. The perpendicular height of the pyramid is `2x`. All measurements are in metres.
 

What is the volume of the pyramid in cubic metres?

  1. `8x^3`
  2. `9x^3`
  3. `12x^3`
  4. `24x^3`
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`A`

Show Worked Solution
`text(Volume)` `= 1/3Ah`
  `= 1/3(4x xx 3x xx 2x)`
  `= 8x^3`

 
`=>A`

Measurement, STD2 M1 2017 HSC 30e

A solid is made up of a sphere sitting partially inside a cone.

The sphere, centre `O`, has a radius of 4 cm and sits 2 cm inside the cone. The solid has a total height of 15 cm. The solid and its cross-section are shown.
 


 

Using the formula  `V=1/3 pi r^2h`  where `r`  is the radius of the cone's circular base and `h` is the perpendicular height of the cone, find the volume of the cone, correct to the nearest cm³?   (3 marks)

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`113\ text{cm}^3`

Show Worked Solution

`V = 1/3 xx text(base of cone × height)`
 

`text(Consider the circular base area of the cone,)`

`text{Using Pythagoras:}`

`x^2` `= 4^2-2^2 = 16-4 = 12`
`x` `= sqrt12\ text(cm)`

 

`:. V` `= 1/3 xx pi xx (sqrt12)^2 xx (15-6)`
  `= 1/3 xx pi xx 12 xx 9`
  `= 113.097…`
  `= 113\ text{cm}^3\ text{(nearest cm}^3 text{)}`

Measurement, STD2 M1 2015 HSC 8 MC

The Louvre Pyramid in Paris has a square base with side length 35 m and a perpendicular height of 22 m.
 

What is the volume of this pyramid, to the nearest m³?

  1. `257\ text(m)^3`
  2. `1027\ text(m)^3`
  3. `8983\ text(m)^3`
  4. `26\ 950\ text(m)^3`
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`C`

Show Worked Solution
`V` `= 1/3Ah`
`A` `= 35 xx 35`
  `= 1225\ text(m)^2`

 

`:.V` `= 1/3 xx 1225 xx 22`
  `= 8983.33…\ text(m)^3`

 
`=>C`

Measurement, STD2 M1 2013 HSC 12 MC

A square pyramid fits exactly on top of a cube to form a solid.

2013 12 mc

What is the volume of the solid?

  1. 513 cm³
  2. 999 cm³
  3. 1242 cm³
  4. 1539 cm³
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`B`

Show Worked Solution
`text(Volume )` `=text{Vol (cube)} +text{Vol (pyramid)}`
  `=l^3+1/3Ah`
  `=(9xx9xx9)+(1/3xx9xx9xx10)`
  `=999\ text(cm)^3`

 
`=>\ B`