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Volume, SM-Bank 152

The 3D shape below is a composite prism consisting of a half-cylinder and a rectangular prism.
 

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

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\( 4881.7\ \text{cm}^3\ (\text{1 d.p.})\)

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\(\text{Diameter of cylinders = 10 cm  }\longrightarrow\  r=5\ \text{cm}\)

\(\text{Volume = rectangular prism + ½ cylinder}\)

\(\therefore\ V\) \(=l\times b\times h+\dfrac{1}{2}\times\pi r^2h\)
  \(=24\times 6.5\times 25+\dfrac{1}{2}\times\pi\times 5^2\times 25\)
  \(=3900+981.747\dots\)
  \(=4881.747\dots\)
  \(\approx 4881.7\ \text{cm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 151

A piece of metal in the shape of a rectangular prism has had two cylindrical holes, each with a diameter of 8 millimetres, drilled through it.
 

Calculate the volume of the remaining metal in cubic millimetres, giving your answer correct to one decimal place.  (2 marks)

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\( 4937.6\ \text{mm}^3\ (\text{1 d.p.})\)

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\(\text{Diameter of cylinders = 8 mm  }\longrightarrow\  r=4\ \text{mm}\)

\(\text{Volume = rectangular prism – 2 × cylinders}\)

\(\therefore\ V\) \(=l\times b\times h-2\times\pi r^2h\)
  \(=32\times 16\times 12-2\times\pi\times 4^2\times 12\)
  \(=6144-1206.371\dots\)
  \(=4937.628\dots\)
  \(\approx 4937.6\ \text{mm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 150

A a skateboard ramp has been constructed using a rectangular prism that has had a quarter-cylinder removed to create the curved surface.
 

Calculate the volume of the skateboard ramp, giving your answer correct to one decimal place.  (2 marks)

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\( 92.2\ \text{m}^3\ (\text{1 d.p.})\)

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\( V\) \(=l\times b\times h-\dfrac{1}{4}\times\pi r^2h\)
  \(=7\times 5\times 6-\dfrac{1}{4}\times\pi\times 5^2\times 6\)
  \(=210-117.809\dots\)
  \(=92.190\dots\)
  \(\approx 92.2\ \text{m}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 149

A a chicken feeder has been constructed using a rectangular prism and a quarter-cylinder.
 

Calculate the volume of the chicken feeder, giving your answer correct to one decimal place.  (2 marks)

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\( 61\ 703.4\ \text{cm}^3\ (\text{1 d.p.})\)

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\(V\) \(=\dfrac{1}{4}\times\pi r^2h+l\times b\times h\)
  \(=\dfrac{1}{4}\times\pi\times 24^2\times 60+24\times 24\times 60\)
  \(=27\ 143.360\dots+34\ 560\)
  \(=61\ 703.360\dots\)
  \(\approx 61\ 703.4\ \text{cm}^3\ (\text{1 d.p.})\)

Volume, SM-Bank 148

A large machinery storage shed has been constructed on a property. The shed is made up of a rectangular prism and a half cylinder.
 

Calculate the volume of the machinery shed, giving your answer to the nearest cubic metre.  (2 marks)

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\(3126\ \text{m}^3\)

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\(\text{Diameter semi-circle = 12 m }\rightarrow\ \text{r = 6 m}\)

\(\therefore\ V\) \(=\dfrac{1}{2}\times\pi r^2h+l\times b\times h\)
  \(=\dfrac{1}{2}\times\pi\times 6^2\times 19+12\times 9\times 19\)
  \(=1074.424\dots+2052\)
  \(=3126.424\dots\)
  \(\approx 3126\ \text{m}^3\ (\text{nearest m}^3)\)

Volume, SM-Bank 147

A concrete half-pipe was constructed in a park. The pipe has a constant thickness 0.5 metres.
 

  1. Calculate the volume of the concrete used to create the half-pipe, giving your answer to the nearest cubic metre.  (2 marks)
  2. Calculate the capacity of the concrete used in kilolitres.  (1 mark)
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a.    \(19\ \text{m}^3\)

b.    \(9\ \text{kL}\)

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a.    \(\text{diameter large semi-circle = 4 m }\rightarrow\ \text{R = 2 m}\)

\(\text{diameter small semi-circle = 3 m }\rightarrow\ \text{r = 1.5 m}\)

\(\therefore\ V\) \(=\dfrac{1}{2}\times\pi R^2h-\dfrac{1}{2}\times\pi r^2h\)
  \(=\dfrac{1}{2}\times\pi\times 2^2\times 7-\dfrac{1}{2}\times\pi\times 1.5^2\times 7\)
  \(=19.242\dots\)
  \(\approx 19\ \text{m}^3\ (\text{nearest m}^3)\)

 

b.    \(\text{1 kL = 1 m}^3\)

\(\therefore\ 19\ \text{m}^3\) \(=19\ \text{kL}\)

Volume, SM-Bank 014

A closed cylindrical water tank has external diameter 3.5 metres.

The external height of the tank is 2.4 metres.

The walls, floor and top of the tank are made of concrete 0.25 m thick.
 

Geometry and Trig, FUR2 2006 VCAA 3

  1. What is the internal radius, \(r\), of the tank?  (1 mark)

  2. What is the internal height, \(h\), of the tank?  (1 mark)

  3. Determine the maximum amount of water this tank can hold.
    Write your answer correct to the nearest cubic metre.  (2 marks) 

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a.    \(1.5\ \text{m}\)

b.    \(13\ \text{m}^3\ \text{(nearest m³)}\)

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a.   \(\text{Internal radius}\ (r)\)

\(=\dfrac{1}{2}\times (3.5-2\times 0.25)\)

\(=1.5\ \text{m}\)
 

b.    \(\text{Internal Height}\ (h)\) \(=2.4-(2\times 0.25)\)
    \(=1.9\ \text{m}\)

 

c.    \(\text{Volume}\) \(=\pi r^2 h\)
    \(=\pi\times 1.5^2\times 1.9\)
    \(=13.430\dots\)
    \(=13\ \text{m}^3\ \text{(nearest m³)}\)