WISKUNDE
GRAAD 12
NOG OEFENINGE
  
Omtrek, oppervlakte en volume : antwoorde
  
MATHEMATICS
GRADE 12
MORE EXERCISES
  
Perimeter, area and volume : answers
  
  
Antwoorde  / Answers  1  
  
     
      1.1  Oppervlakte =  lengte X breedte        
  
                                                       120
               breedte = 120 mm    =  ────  cm
                                                       10
  
                            =  12  cm
  
               Oppervlakte  =  lengte X breedte
  
                                    =  20 X 12  cm2
  
                                    =  240  cm2             [ V/Q 1.1 ]
  
     
      1.2  Oppervlakte =  ½ X basis X hoogte
  
                                 =  ½ X 45 X 31  cm2
  
                                 =  697,5  cm2            [ V/Q 1.2 ]
  
     
      1.3  Oppervlakte =  ½ X basis X hoogte
  
                                 =  ½ X 30 X 5,5  m2
  
                                 =  82,5  m2               [ V/Q 1.3 ]
  
     
      1.4  Oppervlakte =  lengte X hoogte
  
                                 =  18 X 8  mm2
  
                                 =  144  mm2            [ V/Q 1.4 ]
  
     
      1.5  Oppervlakte =  ½ X basis X hoogte
  
                                 =  ½ X 38 X 18  cm2
  
                                 =  342  cm2             [ V/Q 1.5 ]
  
     
      1.6  Oppervlakte =  ½ X basis X hoogte
  
                                 =  ½ X 56 X 46,4  m2
  
                                 =  1 299,200  m2       [ V/Q 1.6 ]
  
     
      1.7  Oppervlakte =  π X radius2 
  
                                 =  π X 23,42  cm2
  
                                 =  1 720,210  cm2      [ V/Q 1.7 ]
  
  
  
      1.8  radius  =  ½  middellyn
  
                         =  ½ X 4,65  m    =  2,325 m
  
             Oppervlakte =  π X radius2 
  
                                 =  π X 2,3252  m2
  
                                 =   16,982  m2          [ V/Q 1.8 ]
  
  
  OF  
  
  
                                        π X middellyn2 
              Oppervlakte =  ──────────
                                                   4
  
                                             π X 4,652 
                                   =  ──────────  m2
                                                   4
                                   =  16,982  m2          [ V/Q 1.8 ]
  
  
      1.1  Area  =  length X breadth    
  
                                                       120
               breadth = 120 mm    =  ────  cm
                                                       10
  
                            =  12  cm
  
               Area  =  length X breadth
  
                         =  20 X 12  cm2
  
                         =  240  cm2                       [ V/Q 1.1 ]
  
     
      1.2  Area =  ½ X base X height
  
                      =  ½ X 45 X 31  cm2
  
                      =  697,5  cm2                       [ V/Q 1.2 ]
  
     
      1.3  Area =  ½ X base X height
  
                      =  ½ X 30 X 5,5  m2
  
                      =  82,5  m2                           [ V/Q 1.3 ]
  
     
      1.4  Area =  length X height
  
                      =  18 X 8  mm2
  
                      =  144  mm2                         [ V/Q 1.4 ]
  
     
      1.5  Area =  ½ X base X height
  
                      =  ½ X 38 X 18  cm2
  
                      =  342  cm2                          [ V/Q 1.5 ]
  
     
      1.6  Area =  ½ X base X height
  
                      =  ½ X 56 X 46,4  m2
  
                      =  1 299,200  m2                  [ V/Q 1.5 ]
  
     
      1.7  Area =  π X radius2 
  
                      =  π X 23,42  cm2
  
                      =  1 720,210  cm2                [ V/Q 1.7 ]
  
  
  
      1.8  radius  =  ½  diameter
  
                         =  ½ X 4,65  m    =  2,325 m
  
             Area =  π X radius2 
  
                      =  π X 2,3252  m2
  
                      =   16,982  m2                      [ V/Q 1.8 ]
  
  
  OR  
  
  
                            π X diameter2 
              Area =  ──────────
                                      4
  
                            π X 4,652 
                      =  ──────────  m2
                                      4
                      =  16,982  m2                       [ V/Q 1.8 ]
  
  
Antwoorde  / Answers  2  
  
       Ons gebruik die volgende afkortings hier :
  
       Lengte : L ; Breedte = B ; Wydte = W ;
  
       hoogte : h; basis : b; sy : s
  
       skuinssy : hs ; Buite-oppervlakte : BO;
  
       Here we use the following abbreviations :
  
       Length : L ; Breadth = B ; Width = W ;
  
       height : h; base : b; side : s
  
       hypotenuse : hs ; Surface area : SA;
  
     
      2.1  Lengte / Length  =  L  =  0,85  m  =  0,85 X (10 X 10) cm     =  85 cm
  
                                                                         185
             Hoogte / Height  =  h  =  185  mm  =  ───  cm    =  18,5 cm
                                                                          10
  
  
             Volume =  L X B X h
  
                          =  85 X 21 X 18,5  cm3
  
                          =  33 022,5 cm3
  
  
             BO / SA  =  2 X (L X B)  +  2 X (L X h)  +  2 X (B X h)
  
                            =  2 X (85 X 21)  +  2 X (85 X 18,5)  +  2 X (21 X 18,5)  cm2
  
                            =  3 570  +  3 145  +  777  cm2
  
                            =  7 492  cm2
  
  
  
  OF / OR  

  
             BO / SA  =  (omtrek van boom / Perimeter of bottom) X h + 2 X (opperv. van kopstuk / area of end)
  
                            =  2(b + h) X L  +  2 X (b X h)
  
                            =  2 X (21 + 18,5) X 85  +  2 X (21 X 18,5)  cm2
  
                            =  6 715  +  777  cm2
  
                            =  7 492  cm2                                                                                                           [ V/Q 2.1 ]
  
     
      2.2   Volume =  s X s X s    =  s3
  
                           =  673  cm3
  
                           =  300 703 cm3
  
  
             BO / SA  =  6 X (oppervlakte van een sy  /  area of one side)    =  6 X s2  
  
                            =  6 X 672  
  
                            =  26 934    cm2                                                                                                         [ V/Q 2.2 ]
  
     
      2.3  Skuinssy / Hypotenuse  =  hs  :  hs2  =  152  +  182  =  549
  
                                                                  hs  =  23,431 cm
  
             basis =  reghoekige driehoek  /  base  =  right-angled triangle
  
             Volume =  oppervlakte van basis X loodregte hoogte  /  Volume  =  area of base X perpendicular height
  
                          =  ( ½ X b X h )  X  L
  
                          =  ( ½ X 15 X 18 )  X  34  cm3
  
                          =  4 590  cm3
  
             BO / SA  =  omtrek van driehoek X L  +  2 X oppervlakte van driehoek  / 
  
                            =  perimeter of triangle X L  +  2 X area of triangle
  
             BO / SA  =  (23,431 + 15 + 18) X 34  +  2 X ( ½ X 15 X 18)  cm2
  
                            =  1 918,654  +  270  cm2
  
                            =  2 188,654    cm2                                                                                                    [ V/Q 2.3 ]
  
  
     
Antwoorde  / Answers  3.1  
  
             Totale oppervlakte  =  Oppervlakte van twee driehoeke  +  oppervlakte van een reghoek
  
             Total area  =  area of two triangles  +  area of one rectangle
  
             Hoogte van die twee driehoeke  =  Totale lengte  −  lengte van reghoek.  /  
  
             Height of the two triangles  =  Total length  −  length of rectangle.
  
             Hoogte van een driehoek / Height of one triangle  =  ½ ( 6 − 3 )  =    1,5  m
  
             Totale oppervlakte / Total area  =  2 X ( ½ X b X h )  +  L X b
  
                                                               =  2 X ( ½ X 3 X 1,5 )  +  3 X 3  m2
  
                                                               =  13,5  m2                                                                              [ V/Q 3.1 ]
  
  
Antwoorde  / Answers  3.2  
  
             Totale oppervlakte  =  Oppervlakte van twee halwe sirkels  +  oppervlakte van een reghoek
  
             Total area  =  area of two semi-circles  +  area of one rectangle
  
             Total area  =  area of two semi-circles  +  area of one rectangle
  
             Totale opervlakte / Total area   opp. van 1 sirkel / area of 1 circle  +  opp. van reghoek / area of rectangle
  
             Deursnit van sirkel / Diameter of circle  =  D  =  2 m
  
             Radius van sirkel /  Radius of circle  =  ½ X D    =  ½ X 2   =  1 m
  
             Totale opervlakte / Total area   =  π r2 +  LB
  
                                                              =  π X 12 +  3 X 3  m2
  
                                                              =  12,142  m2
  
 
  OF / OR  
 
  
                                                                     π X D2
             Totale opervlakte / Total area   =  ──────  +  LB
                                                                         4
                                                                     π X 22
                                                              =  ──────  +  3 X 3  m2
                                                                         4
                                                              =  12,142  m2                                                                          [ V/Q 3.2 ]
  
  
  
             Totale omtrek  Total perimeter   =  Beweeg een maal op die buitelyne van die figuur om die figuur.  /
  
             Total perimeter   =  Move once around the figure following the exterior lines.
  
             Totale omtrek / Total perimeter   =  Omtrek van twee halwe sirkels + twee 4 m sye van reghoek
  
             Totale omtrek / Total perimeter   =  Perimeter of two semi-circles + two 4 m sides of the rectangle.
  
             Totale omtrek / Total perimeter   =  ½ X π X D  +  4  +  ½ X π X D  +  4
  
                                                                =  ½ X π X 2  +  4  +  ½ X π X 2  +  4  m
  
                                                                =  14,283  m                                                                        [ V/Q 3.2 ]
  
  
Antwoorde  / Answers  3.3  
  
             Totale oppervlakte  =  Oppervlakte van 'n driehoek  +  oppervlakte van 'n reghoek
  
             Total area  =  area of a triangle  +  area of a rectangle
  
             Totale oppervlakte / Total area  =  ½ X b X h  +  L X b
  
                                                               =  ½ X 20 X 5,5  +  20 X 14  m2
  
                                                               =  335  m2                                                                                [ V/Q 3.3 ]
  
  
  
  
Antwoorde  / Answers  3.4  
  
      3.4  Totale oppervlakte  =  Oppervlakte van twee reghoekige driehoeke
  
             Total area  =  area of two right-angled triangles
  
             Ons moet die lengte van sy AC van ΔADC bereken. AC is ook die skuinssy van ΔABC.
  
             We have to calculate the length of side AC of ΔADC. AC is also the hypotenuse of ΔABC.
  
             In ΔABC  :  AC2  =  AB2  +  BC2      .  . .  Pythagoras 
  
                                         =  2002  +  1502
  
                                         =  62 500
  
                                AC  =  √62500    =  250
  
             Totale oppervlakte / Total area  =  ½ X b X h  +  ½ X b X h
  
                                                               =  ½ X 200 X 150  +  ½ X 600 X 250  m2
  
                                                               =  90 000  m2                                                                                [ V/Q 3.4 ]
  
  
  
             Omtrek / Perimeter  =  AB  +  BC  +  CD  +  DA
  
             Ons moet CD se lengte bepaal. CD is die skuinssy van ΔACD / 
  
             We have to calculate the length of side CD of ΔADC. CD is also the hypotenuse of ΔACD.
  
             In ΔACD  :  CD2  =  AD2  +  BC2      .  . .  Pythagoras 
  
                                         =  6002  +  62 5002
  
                                         =  422 500
  
                                CD  =  √422 500    =  650
  
             Omtrek / Perimeter  =  AB  +  BC  +  CD  +  DA
  
                                             =  200  +  150  +  650  +  600  m
  
                                             =  1 600  m                                                                                               [ V/Q 3.4 ]
  
  
Antwoorde  / Answers  3.5  
  
             Totale oppervlakte  =  Oppervlakte van 'n halfsirkel + oppervlakte van 'n reghoekige driehoek  /
  
             Total area  =  area of a semi-circle + area of a right-angled triangle.
  
             Ons moet die lengte van sy AB van ΔABC bereken. AB is 'n reghoeksy van ΔABC.
  
             We have to calculate the length of side AB of ΔABC. AB is a right-angled side of ΔABC.
  
             In ΔABC  :  BC2  =  AB2  +  AC2      .  . .  Pythagoras 
  
                                 132  =  AB2  +  52
  
                                 169  =  AB2  +  25
  
                                 AB2  =  169  −  25
  
                                         =  144
  
                                AC  =  √144    =  12
  
                                D  =  12   of / or   r =  6
  
                                                                     π X D2
             Totale opervlakte / Total area   =  ──────  +  ½ X b X h
                                                                         4
                                                                              π X 122
                                                              =  ½  X  ──────  +   ½ X 5 X 12  m2
                                                                                 4
  
                                                              =  86,549  m2                                                                           [ V/Q 3.5 ]
  
  
  
 
  OF / OR  
 
  
             Totale oppervlakte / Total area  =  ½ X π X r2  +  ½ X b X h
  
                                                               =  ½ X π X 62  +  ½ X 12 X 5  m2
  
                                                               =  86,549  m2                                                                         [ V/Q 3.5 ]
  
  
  
             Omtrek / Perimeter  =  omtrek van halwe sirkel / perimeter of a semi-circle  +  BC  +  CA
  
             Omtrek / Perimeter  =  ½ X π X D  +  BC  +  CA
  
                                             =  ½ X π X 12  +  13  +  5  m
  
                                             =  3,850  m                                                                                               [ V/Q 3.5 ]
  
  
Antwoorde  / Answers  4.  
  
  
       4.1  Oppervlakte van grond / Area of soil  =  Oppervlakte van reghoek / Area of a rectangle    =  LB
  
              Oppervlakte van grond / Area of soil  =  15 X 20  m2   
                                                                         =  300  m2                                                                     [ V/Q 4.1 ]
  
  
       4.2  Nee, die plastic is te naby aan die grond langs die kante en daardie oppervlakte kan nie bewerk word nie.
  
               No, the plastic is too close to the ground along the sides and that area can not be used.
  
                                                                                                                                                              [ V/Q 4.2 ]
  
  
       4.3  Volume lug  /  Volume of air  =  Volume van 'n halwe silinder  /  Volume of a half cylinder
  
                                                                        π D2
                                                             =  ½  ────  X  L
                                                                          4
  
                                                                        π 152
                                                             =  ½  ────  X  20  m3
                                                                          4
  
                                                             =  1 767,146  m3                                                                      [ V/Q 4.3 ]
  
  
Antwoorde  / Answers  5.  
  
  
                                                                           5 000
       5.1  Volume van tenk / Volume of tank  =  ─────        |  1000 liter  =   1 m3
  
                                                                           1 000
  
                                                                     =  5 m3                                                                                [ V/Q 5.1 ]
  
  
       5.2  Reghoekige prisma / Rectangular prism  :
  
              Volume  =  LBh      en / and  V = 5 m3
  
              L  =  2 X B      |  lengte is tweemaal breedte  /  length is double its breadth
  
                  2B X B X 2  =  5
  
                                 5
                    B2  =  ───
                                 4
  
                    B  =  1,118      en dus  / and therefore  L  =  2,236
  
                    Dus afmetings is :  /  Dimensions are  :  L = 2,236 m ,  B = 1,118 m  en / and  h = 2 m        [ V/Q 5.2 ]
  
              Kubus  /  Cube  :    Volume  =  s3
  
                 s3  =  5
                           3
                   s  =  √ 5      =  1,710
  
                    Dus afmetings is :  /  Dimensions are  :  sy / side = 1,170 m                                              [ V/Q 5.2 ]
  
                                                                      π X D2 X h
              Silinder  /  Cylinder  :    Volume  =  ───────
                                                                             4
  
                                             π X D2 X h
                                             ───────  =  Volume
                                                     4
  
                                             π X D2 X 2
                                             ───────  =  5
                                                     4
  
  
                                             π X D2  =  10
  
                                                    D2  =  3,183098862
  
                                                     D  =  √3,183098862    =  1,784
  
                    Dus afmetings is :  /  Dimensions are  :  D  = 1,784 m    en / and    h  =  2 m                     [ V/Q 5.2 ]
  
  
Antwoorde  / Answers  6.  
  
                                                                                                                            1,65
        Lengte van 'n derde van die rak / Length of one third of the shelf  = L  =  ────      =   0,55 m    =  55 cm
                                                                                                                               3
  
       Rakwydte / Width of shelf  =  W  =  25 cm ;    Hoogte tussen rakke / Height between shelves  =RSH  =  23 cm
  
       Pilchards  :
                           D  =  deursnit van blik / diameter of tin  =  7,5 cm
  
                           H  =  hoogte van blik / height of tin  =  110 mm = 11 cm
  
                                                                                                                    L             55
                           Aantal blikke in een ry / Number of tins in one row  =   ───    =  ───    =  7,333 
                                                                                                                    D            7,5
  
                               Daar kan 7 blikke in 1 ry gepsk word.  /  7 tins can be palced in one row.
  
                                                                                                                    W             25
                           Aantal rye op die rak / Number of rows on the shelf  =   ───    =  ───    =  3,333
                                                                                                                    D            7,5
  
                               Daar kan 3 rye op die rak gepak word.  /  3 rows can be packed on the shelf.
  
                                                                                                                                   RSH        23 cm
                           Aantal blikke op mekaar / Number of tins on top of each other  =  ───    =  ────   =  2,091
                                                                                                                                      H          11 cm
  
                               Daar kan 2 blikke op mekaar gepak word.  /  2 tins can be placed on top of one another.
  
                           Aantal blikke in een laag  =  7 blikke in 'n ry  X  3  rye  =  21
  
                           Number of tins in one layer  =  7 tine per row  X  3  rows  =  21
  
                           Total aantal blikke op die rak  =  21 blikke per laag  X  2  lae  =  42
  
                           Total number of tins on the shelf  =  21 tins per layer  X  2  layers  =  42
  
       Tuna  :
                           d  =  deursnit van blik / diameter of tin  =  2r  =  2 X 43 mm    =  86 mm    =  8,6 cm
  
                           h  =  hoogte van blik / height of tin  =  37 mm = 3,7 cm
  
                                                                                                                    L             55
                           Aantal blikke in een ry / Number of tins in one row  =   ───    =  ───    =  6,395 
                                                                                                                    D            8,6
  
                               Daar kan 6 blikke in 1 ry gepsk word.  /  6 tins can be palced in one row.
  
                                                                                                                    W             25
                           Aantal rye op die rak / Number of rows on the shelf  =   ───    =  ───    =  2,907
                                                                                                                    D            8,6
  
                               Daar kan 2 rye op die rak gepak word.  /  2 rows can be packed on the shelf.
  
                                                                                                                                   RSH        23 cm
                           Aantal blikke op mekaar / Number of tins on top of each other  =  ───    =  ────   =  6,216
                                                                                                                                      H          3,7 cm
  
                               Daar kan 6 blikke op mekaar gepak word.  /  6 tins can be placed on top of one another.
  
                           Aantal blikke in een laag  =  6 blikke in 'n ry  X  2  rye  =  12
  
                           Number of tins in one layer  =  6 tine per row  X  2  rows  =  12
  
                           Total aantal blikke op die rak  =  12 blikke per laag  X  6  lae  =  72
  
                           Total number of tins on the shelf  =  12 tins per layer  X  6  layers  =  72
  
                           Jack kan 42 blikkies Pilchards, 72 blikkies Tune in Water en 72 blikkies Tuna in Oil op die rak pak.
  
                           Jack can place 42 tins of Pilchards, 72 tins of Tuna in Water and 72 tins of Tuna in Oil on the shelf
  
                                                                                                                                                                  [ V/Q 6. ]
  
  
Antwoorde  / Answers  7.  
  
     
               Lengte van bakkie / Length of bakkie  = L  =  3 X 100    =   300 cm
  
               Breedte van bakkie / Breadth of bakkie  = B  =  1,75 X 100    =   175 cm
  
               Diepte/hoogte van bakkie / Deapth/Height of bakkie  = H  =  400 ÷ 10    =   40 cm
  
                                                                                               600
               Lengte van sakkie / Length of pocket  = l  =  ────      =   60 cm
                                                                                                10
  
                                                                                                  320
               Breedte van sakkie / Breadth of pocket  = l  =  ────      =   32 cm
                                                                                                   10
  
               Diepte/hoogte van sakkie / Deapth/Height of pocket  = h  =  130 ÷ 10    =   13 cm
  
  
                                                                                                                      300
      7.1   Aantal sakkies / Number of pockets  =    ────      =   5               [ V/Q 7. ]
                                                                                                                       60
  
  
     
                                                                                                                            175
      7.2   Aantal sakkies / Number of pockets  =    ────      =   5,469  =  5        [ V/Q 7. ]
                                                                                                                             32
                                                                                                                                  23;           [ V/Q 7. ]
  
  
     
      7.3   Aantal sakkies in een laag / Number of pockets in one layer  =  5 X 5   =  25                        [ V/Q 7. ]
  
     
                                                                                                                                175
      7.4   Aantal sakkies agter mekaar / Number of pockets across bakkie  =    ────      =   2,917  =  2
                                                                                                                                 60
  
                                                                                                                              300
              Aantal sakkies oor lengte / Number of pockets lengthwise  =    ────      =   9,375  =  9
                                                                                                                               32
  
              Aantal sakkies in een laag / Number of pockets in one layer  =  2 X 9   =  18              [ V/Q 7. ]
  
  
     
                                                                                                     40
      7.5   Aantal sakkies op mekaar / Number of layers  =    ────      =   3,077  =  3
                                                                                                      13
  
               Aantal lae / Number of layers  =  3
  
               Maksimum aantal sakkies  =  aantal per laag  X  aantal lae  =  25  X  3  =  75
  
               Maximum number of pockets  =  number per layer  X  number of layers  =  25  X  3  =  75
  
                                                                                                                                                    [ V/Q 7. ]
  
  
     
      7.6   Massa van 75 sakkies / Mass of 75 pockets  =  75 X 50  =  3 750 kg
  
              Maksimum vrag / Maximum load  =  2 000 kg  =  40 sakkies / pockets
  
              Nee, die vrag is te swaar. / No the load is too great.                                         56             [ V/Q 7. ]
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
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