Volume

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)

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`2.85\ text(m)`

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`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-016

The truncated cone, pictured below, is made by cutting a right cone of height 60 centimetres.

Find the volume of the truncated cone, giving your answer to the nearest cubic centimetre. (4 marks)

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`7916\ text(cm)^3`

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`text{Radius (original cone) = 12 cm}`

`text{Radius (small cone) = 6 cm}`

`text{Volume}`	`=\ text{Original cone – Small cone}`
	`=(1/3 xx pi xx 12^2 xx 60)-(1/3 xx pi xx 6^2 xx 30)`
	`= 7916.81…`
	`=7916\ text{cm}^3\ \ text{(nearest cm}^3 text{)}`

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)

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`1842.94\ text(cm)^3`

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`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)

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`150\ text(m)^3`

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`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)

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

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

Volume, SMB-012

The solid below is made from joining a rectangular prism to a triangular prism, with all measurements in centimetres.

Find the volume of the solid, giving your answer to the nearest cubic centimetre. (3 marks)

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`11\ 451\ text(cm)^3`

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`text{Area (face)}`	`= (9 xx 13) + (1/2 xx 13 xx 7)`
	`=162.5\ text{cm}^2`

`text(Volume)`	`= A h`
	`= 162.5 xx 6`
	`= 975\ text(cm)^3`

Volume, SMB-011

The cylinder shown below has a diameter of 18 centimetres and a length of 45 centimetres.

Find the volume of the cylinder, giving your answer to the nearest cubic centimetre. (2 marks)

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`11\ 451\ text(cm)^3`

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`text(Volume)`	`= pi r^2 h`
	`= pi xx 9^2 xx 45`
	`= 11\ 451.10…`
	`= 11\ 451\ text(cm)^3\ \ text{(nearest cm}^3 text{)}`

Volume, SMB-010

The solid below is a triangular prism. All measurements are in centimetres.

Find the volume of the solid, in cubic centimetres. (2 marks)

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`378\ \text{cm}^3`

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`text{Area (triangle)}`	`=1/2 xx b xx h`
	`=1/2 xx 9 xx 12`
	`=54\ text{cm}^2`

`text(Volume)`	`= A xx h`
	`= 54 xx 7`
	`= 378\ \text{cm}^3`

Volume, SMB-009

An ancient temple, modelled below, was constructed as a trapezoidal prism.

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

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`432\ \text{m}^3`

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`text{Area (trapezium)}`	`=1/2h(a+b)`
	`=1/2 xx 4 (10+14)`
	`=48\ text{m}^2`

`text(Volume)`	`= A xx h`
	`= 48 xx 9`
	`= 432\ \text{m}^3`

Volume, SMB-008

A jewellery box in the shape of a rectangular prism is pictured below with all measurements in centimetres.

Find the volume of the jewellery box, giving your answer correct to the nearest cubic centimetre. (2 marks)

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`271\ text{cm}^3`

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`text(Volume)`	`= l xx b xx h`
	`= 9.5 xx 4.2 xx 6.8`
	`= 271.32`
	`= 271\ text{cm}^3\ text{(nearest cm}^3 text{)}`

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)

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`6370.6\ text{cm}^3`

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`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)

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`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)

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`2714\ text{cm³}`

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`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}`

Volume, SMB-004

Calculate the volume of earth that must be excavated so that the pool shell below fits exactly into the hole. All measurements are in metres. (3 marks)

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\(136.8\ \text{m}^3\)

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\(\text{The face of the pool is a rectangle and trapezium (composite).}\)

\(\text{Area (face)}\)	\(= (2.3 \times 4) + \dfrac{1}{2} h (a+b)\)
	\(= 9.2 + \dfrac{1}{2} \times 8 (2.3+1.1) \)
	\(=22.8\ \text{m}^2\)

\(\text{Volume}\)	\(=Ah\)
	\(=22.8 \times 6\)
	\(=136.8\ \text{m}^3\)

Volume, SMB-003

Steel rods are manufactured in the shape of equilateral triangular prisms.

Find the volume of the prism (answer correct to 1 decimal place). (3 marks)

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`3464.1\ text(cm)^3`

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`text{Area of triangular face (using sine rule)}`

`=1/2 xx 10 xx 10 xx sin60°`

`=43.301…`

`text(Volume)`	`=Ah`
	`=43.301… xx 80`
	`=3464.10…`
	`=3464.1\ text{cm³ (1 d.p.)}`

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)

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`5.4\ text(m)^3`

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`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`

Measurement, STD2 M1 2022 HSC 28

A dam is in the shape of a triangular prism which is 50 m long, as shown.

Both ends of the dam, `A B C` and `D E F`, are isosceles triangles with equal sides of length 25 metres. The included angles `B A C` and `E D F` are each `150^@`.

Calculate the number of litres of water the dam will hold when full. (4 marks)

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`7\ 812\ 500\ text{L}`

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`V=Ah`

`text{Use sine rule to find}\ A:`

`A`	`=1/2 ab\ sinC`
	`=1/2 xx 25 xx 25 xx sin150^@`
	`=156.25\ text{m}^2`

`:.V`	`=156.25 xx 50`
	`=7812.5\ text{m}^3`

`text{S}text{ince 1 m³ = 1000 litres:}`

`text{Dam capacity}`	`=7812.5 xx 1000`
	`=7\ 812\ 500\ text{L}`

♦♦ Mean mark 33%.

Measurement, STD2 M1 2018 HSC 30a

A cylindrical water tank has a radius of 9 metres and a capacity of 1.26 megalitres.

What is the height of the water tank? Give your answer in metres, correct to two decimal places. (3 marks)

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`4.95\ text{m}`

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`text{Converting megalitres to m³ (using 1 m³ = 1000 L):}`

♦ Mean mark 48%.

`1.26\ text(ML)`	`= (1.26 xx 10^6)/(10^3)`
	`= 1.26 xx 10^3\ text(m)^3`
	`= 1260\ text(m)^3`

`V`	`= pir^2h`
`1260`	`= pi xx 9^2 xx h`
`h`	`= 1260/(pi xx 9^2)`
	`= 4.951…`
	`= 4.95\ text{m (2 d.p.)}`

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`

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

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

`text(Find)\ x\ \ 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 2016 HSC 12 MC

A container is in the shape of a triangular prism which has a capacity of 12 litres. The area of the base is 240 cm².

What is the distance, `h`, between the two triangular ends of the container?

5 cm
20 cm
25 cm
50 cm

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`=> D`

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♦♦ Mean mark 35%.

`text{1 mL = 1 cm}^3\ \ =>\ \ text{1 L = 1000 cm}^3`

`text(Volume)`	`= Ah`
`12\ 000`	`= 240 xx h`
`h`	`= (12\ 000)/240`
	`= 50\ text(cm)`

`=> D`

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?

`257\ text(m)^3`
`1027\ text(m)^3`
`8983\ text(m)^3`
`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 2005 HSC 23b

A clay brick is made in the shape of a rectangular prism with dimensions as shown.

Calculate the volume of the clay brick. (1 mark)

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Three identical cylindrical holes are made through the brick as shown. Each hole has a radius of 1.4 cm.

What is the volume of clay remaining in the brick after the holes have been made? (Give your answer to the nearest cubic centimetre.) (3 marks)

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What percentage of clay is removed by making the holes through the brick? (Give your answer correct to one decimal place.) (1 mark)

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`text(1512 cm)^3`
`text{1364 cm}^3`
`text{9.8%}`

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i.	`V`	`= l × b × h`
		`= 21 × 8 × 9`
		`= 1512\ text(cm)^3`

ii. `text(Volume of each hole)`

`= pir^2h`

`= pi × 1.4^2 × 8`

`= 49.260…\ text(cm)^3`

`:.\ text(Volume of clay still in brick)`

`= 1512 − (3 × 49.260…)`

`= 1364.219…`

`= 1364\ text{cm}^3\ text{(nearest whole)}`

iii. `text(Percentage of clay removed)`

`= ((3 × 49.260…))/1512 × 100`

`= 9.773…`

`= 9.8 text{% (1 d.p.)}`

Measurement, STD2 M1 2014 HSC 27c

The base of a water tank is in the shape of a rectangle with a semicircle at each end, as shown.

The tank is 1400 mm long, 560 mm wide, and has a height of 810 mm.

What is the capacity of the tank, to the nearest litre? (4 marks)

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`581\ text(L)`

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`V = Ah`

♦ Mean mark 41%
STRATEGY: Adjusting measurements to metres makes the final conversion to litres simple.

`text(Finding Area of base)`

`text(Semi-circles have radius 280 mm) = 0.28\ text(m)`

`:.\ text(Area of 2 semicircles)`

`=2 xx 1/2 xx pi r^2`

`= pi xx (0.28)^2`

`= 0.2463…\ text(m)^2`

`text(Area of rectangle)`

`= l xx b`

`= (1.4-2 xx 0.28) xx 0.56`

`= 0.4704\ text(m)^2`

`:.\ text(Volume)`	`= Ah`
	`= (0.2463… + 0.4704) xx 0.810`
	`= 0.580527…\ text(m)^3`
	`= 580.527…\ text(L)\ \ text{(using 1m³} = 1000\ text{L)}`
	`= 581\ text(L)\ text{(nearest L)}`

Measurement, STD2 M1 2012 HSC 6 MC

What is the volume of this rectangular prism in cubic centimetres?

6 cm³
600 cm³
60 000 cm³
6 000 000 cm³

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`C`

Show Worked Solution

`text{Convert all measurements to centimetres:}`

`V`	`= l xx b xx h`
	`=30 xx 5 xx 400`
	`=60\ 000\ \ text(cm)^3`

`=> C`

`A`	`=1/2 h(a+b)`
	`=1/2xx8(11+5)`
	`=64\ text(cm²)`