text: You are on page 1of 4
  
  # Further X2 Volumes Problems
  
  (a)
  
  ## Figure 1 below shows a scale model of the volcano Mt Rekrap.
  
  The base of the model is elliptical in shape with the axes 60 cm by 40 cm reducing
  uniformly to a circle of radius 12 cm at the top.
  The hollow core of the model has circular cross sections with a circle of radius 6 cm at the
  base rising uniformly to a circle also of radius 12 cm at the top.
  The model is 24 cm high.
  Figure 2 shows the top view of the cross sectional area of the volcano.
  
  Figure 1
  
  24 cm
  h cm
  
  40 cm
  
  60 cm
  Figure 2
  
  3
  h
  4
  
  (i)
  
  ## Show that at height h, the length of the semi-major axis is given by a = 30
  
  (ii)
  
  Show that the area of the cross sectional slice at height h is given by
  
  A=
  9024 448h + 3h 2
  16
  You may assume that the area of an ellipse with semi-major axis a and semi-minor
  axis b is given by ab .
  
  (iii)
  
  ## Find the volume of the scale model of Mt Rekrap.
  
  2
  4
  
  2
  [6336]
  
  7
  7
  QUESTION 5. (Continued)
  QUESTION 5. (Continued)
  HSC 1998
  (b)
  (b)
  
  Marks
  Marks
  9
  
  a
  a
  
  Z
  Z
  
  A
  A
  
  a
  a
  
  Y
  Y
  
  W
  W
  
  C
  C
  
  R
  R
  
  S
  S
  
  x
  x
  
  P
  P
  
  Q
  Q
  x
  x
  
  X
  X
  
  b
  b
  
  B
  B
  
  The diagram shows a sandstone solid with rectangular base ABQP of length
  The diagram shows a sandstone solid with rectangular base ABQP of length
  b metres and width a metres. The end PQRS is a square, and the other end ABC
  b metres and width a metres. The end PQRS is a square, and the other end ABC
  is an equilateral triangle. Both ends are perpendicular to the base.
  is an equilateral triangle. Both ends are perpendicular to the base.
  Consider the slice of the solid with face WXYZ and thickness x metres, as
  Consider the slice of the solid with face WXYZ and thickness x metres, as
  shown in the diagram. The slice is parallel to the ends and AW = BX = x metres.
  shown in the diagram. The slice is parallel to the ends and AW = BX = x metres.
  (i) Find the height of the equilateral triangle ABC.
  (i) Find the height of the equilateral triangle ABC.
  (ii) Given that the triangles CRS and CYZ are similar, find YZ in terms of a,
  (ii) Given that the triangles CRS and CYZ are similar, find YZ in terms of a,
  b and x .
  b and x .
  (iii) Let the perpendicular height of the trapezium WXYZ be h metres. Show
  (iii) Let the perpendicular height of the trapezium WXYZ be h metres. Show
  that
  that
  
  ((
  
  ))
  
  a
  x
  h = a 3 + 2 3 x .
  h = 2 3 + 2 3 b .
  2
  b
  (iv) Hence show that the cross-sectional area of WXYZ is given by
  (iv) Hence show that the cross-sectional area of WXYZ is given by
  
  [[((
  
  ]]
  
  a22
  a
  2 3 x + b 3 (b + x ) .
  4 b22 2 3 x + b 3 (b + x ) .
  4b
  (v) Find the volume of the solid.
  (v) Find the volume of the solid.
  
  ))
  
  9
  9
  
  SGS 6A q23
  
  ## (b) The region bounded by arc AB and the chord AB is rotated
  
  about the chord AB. Show that the solid formed has volume
  103
  r 3 cubic units.
  6 2
  
  EXTENS ION
  
  ## 23. [A First Principles Approach] The region bounded by y = x2
  
  and y = x is rotated about the line y = x to generate a solid.
  In this question you will find the volume of this solid by first
  CHAPTER 6: Volumes
  approximating a small portion with a disk.
  
  y ' x2
  y'x
  
  "h
  r
  
  &
  Let P (x, x2) be a typical point on the parabola. From x to x + x
  P#x\$ x %
  (b) 4
  we will approximate the area between the curve and the line16(a)
  with
  y
  x3
  "x
  1
  a rectangle. One corner of this rectangle is at P and a side is on
  the line y Exercise
  = x. Let6A
  the(Page
  dimensions
  27) of this rectangle be r h as
  "
  shown in the
  diagram.
  !)#
  )#
  1(a)
  (ii) 3
  (b)(i) 9
  y
  14
  5
  x
  (a) Show that h (ii)
  = 2 x and find r as a function of x.
  !" "
  #
  (b) When the rectangle is rotated about the line y = x it generates a cylindrical prism
  with volume V ."Find V
  & .
  
  Chapter Six
  
  y\$x %1
  
  ## (d) Hence show that
  
  !" the volume
  " (i) x of the solid is
  2(a)
  
  30 2
  
  (b) 8
  15
  
  6B The Method
  ofy \$Cylindrical
  Shells
  x
  #
  #
  
  ## Inevitably when" tackling yvolume
  
  \$'#'! x problems there will be
  SGS
  2012
  THSC q16
  b result
  certain
  questions
  which
  in awkward integrals. As
  an example, consider the
  solid
  generated
  when the region
  x
  "
  #
  2
  between y = 4x x and y = x is rotated about the
  y-axis, as shown on the right. The volume
  can be found
  32
  (b)(i) 5
  3(a)
  (ii) 8
  by the methods ofySection 6A since each
  cross-section is
  (ii)
  an annulus. However the situation is complicated by the
  (
  fact that the formula for the outer radius changes. Below
  ! is router = y whilst above y = 3 it is
  router = 2 + 4 y. Hence
  (i) the integral must be split
  into two parts. (Try this as an exercise.)
  
  ## Exercise 6B (Page 32)
  
  1(a) y
  
  (b)
  
  y
  router
  (
  
  )
  (
  
  router
  ( )
  2(a)
  
  #x x
  (b) 128
  5
  
  4(a)
  
  2882
  2
  2 2
  2
  19(a) 43 ab
  (b) 2 ab
  (c) 2 abc
  3
  9
  16
  92
  20(a) AD = 5 , CD = 5
  (c) 15 cm
  1
  2
  
  2 2
  23(a) (x x ) (b) V = (x x ) x
  2
  2
  
  2 2
  (c) V = (x x )
  2
  17(b)
  
  (b) 128
  5
  
  y
  Chapter 6: Volumes
  
  volumes
  
  5/4/14
  
  c 2014
  
  3(a)
  
  128
  3888
  (b)
  3
  5
  3
  9
  (b)
  5
  14
  8
  (b) 8
  3
  256
  (b) 192
  15
  5
  8
  (b) 8
  3
  
  (c) 10
  
  The diagram above shows (the parabola y = 3 + 2x x2 and its4(a)reflection in the y-axis.
  5(a) same height y when rotated about
  The vertical strips shown will generate cylindrical shells of the
  6(a)
  the line x = 1.
  6
  5 7
  7(a)shells is 4 y
  (i)
  Show that the sum
  of the
  areas of these two cylindrical
  1
  48
  7
  2
  4
  
  6(a)
  
  (ii)
  
  (b)
  
  (c) 10
  
  9(a)
  
  (b) 16
  3
  
  7
  Find
  the volume of the solid formed when the region bounded* by the parabolas
  8 32
  3 x-axis is rotated about the line x = 1.
  and the
  9 256
  15
  44
  
  10(a) 8
  (b) 224
  (c) 8
  c.u.
  3
  15
  
  "
  3
  
  11(a) 9
  (b) 15
  14
  7
  8
  (b) 8
  3
  296
  13(b) 15
  32a3
  112a3
  14(a)
  (b)
  15
  15
  15(b) 8
  3
  
  12(a)
  
  !&
  (c)
  
  128a3
  15
  
  "
  
  &
  
  41
  
  Marks
  Question 8 (continued)
  HSC 2002 q8
  38
  
  CHAPTER 6: Volumes
  
  (b)
  
  ## 12. In the diagram on the rightKis a cylindrical wedge ABCD. The N
  
  height of the cylinder is equal to the diameter of its base. Let the
  L
  M
  radius of the base be r units.
  (a) Show that the typical triangular cross-section shaded has area
  (r 2 x2 ) square units.
  
  (b) Hence show that the wedge has volume 43 r 3 cubic units.
  
  A
  (x, y)
  
  B
  !r
  !r
  
  ## 13. The diagram on the right shows a cylindrical drinking glass of
  
  B glass
  interior radius 4 cm and perpendicular height 12 cm. The
  a
  is filled with water which is then drunk slowly until half of the
  bottom of the glass is exposed. Use the methods
  of the previous
  F
  question to find the volume of water aremaining.
  
  y
  r
  C
  r
  x
  
  m
  12 c
  
  4 cm
  
  A
  In the diagram,
  ABshows
  and CD
  are line
  length Its
  2a in horizontal
  4 m planes
  14. The diagram
  to the right
  a large
  tanksegments
  of depth 6ofmetres.
  apart. The
  of CD is
  vertically
  at aits
  distance
  base and
  top are2asquares
  withmidpoint
  sides of 2E metres
  and
  4 metresabove the midpoint F
  of
  AB,
  and
  AB
  lies
  in
  the
  SouthNorth
  direction,
  while
  CD lies in the WestEast
  respectively. A typical square cross-section of side length x metres
  direction.
  is shown
  h metres above the base.
  6m
  (a) Show that x =
  
  + 2.
  
  ## The rectangle3KLMN is the horizontal cross-section of the tetrahedron ABCDhat
  
  (b) Hence
  find xthe
  capacity
  of the tank.
  distance
  from
  the midpoint
  P of EF (so PE = PF = a).
  2m
  
  (i)
  
  ## 15. A rubbish skip
  
  onofa the
  building
  site KLMN.
  has a rectangular base 6 metres by 3 metres, and its
  area
  rectangle
  perpendicular height is 2 metres. Its sides are trapezia that slope outwards from bottom
  to top. The open top is a rectangle 7 metres by 4 metres.
  (ii) Find the volume of the tetrahedron ABCD.
  !
  "!
  "
  (a) Show that a rectangular cross-section h metres above the base has area h2 + 6 h2 + 3
  square metres.
  
  SGS
  6C q16
  (b) Hence
  find the capacity of the skip.
  
  4
  2
  
  End of paper
  
  16.
  
  C
  Q
  
  P
  U
  
  T
  A
  
  CD
  
  O
  S
  
  AB
  R
  
  The first diagram shows two identical intersecting cylinders of radius r. The second
  diagram is the view from above. Their axes AB and CD intersect at 90 at the point O.
  The third diagram shows the solid which is common to both cylinders, bounded at its
  widest point by the horizontal square P QRS.
  (a) The typical square cross-section T U V W shown is parallel to the square P QRS and
  y units above it. Find an expression for the area of this typical cross-section of the
  42
  CHAPTER 6: Volumes
  solid in terms of y.
  
  SGS NOTES
  
  14
  
  (b) Hence find the volume that is common to the two intersecting cylinders.
  10(a)
  
  (b) 27
  2
  
  y
  #
  
  ## SGS Mathematics Year 12
  
  Chapter 6: Volumes
  
  volumes
  "
  
  5/4/14
  
  c 2014
  
  45 23 m3
  2
  2
  3
  16(a) 4(r y ) (b) 16
  3 r
  2
  2
  4 3
  18(a) a x
  (b) 3 a
  
  1 2
  19(a) 23 a (b) ax
  (e) 12
  a b(5 + 2 3) m
  b2
  
  R
  20(a)(ii) (1 e
  ) (c)(i) R 2 (ii) (1
  2
  R2
  (d) (1 e
  ) 4I 2 (1 e2R ) (e
  15(b)
metadata: {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}