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Help me answer this maths question. Very urgent

Discussion in 'General Chat' started by sheetal2009, Feb 17, 2012.

0
  1. #1
    An ant lives on the surface of a cube with edges of length 7cm. It is currently
    located on an edge x cm from one of its ends. While traveling on the surface of the cube,
    it has to reach the grain located on the opposite edge (also at a distance xcm from one
    of its ends) as shown below.
    (i) What is the length of the shortest route to the grain if x = 2cm? How many routes of
    this length are there?
    (ii) Find an x for which there are four distinct shortest length routes to the grain

    image.png

    Please see attach image for figure. Please tell me the steps you have followed to arrive at the solution.
     
    sheetal2009, Feb 17, 2012 IP
  2. digitalpoint

    digitalpoint Overlord of no one Staff

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    #2
    18cm. I arrived at the solution by using my brain.
     
    digitalpoint, Feb 17, 2012 IP
  3. sarahk

    sarahk iTamer Staff

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    #3
    Not exactly hard. I'll get my 12 and 14 year olds to do your homework for you, shall I?
     
    sarahk, Feb 17, 2012 IP
  4. sheetal2009

    sheetal2009 Well-Known Member

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    #4
    They have also given me an example

    Questions on three dimensional geometry sometimes require the student to consider a
    two-dimensional representation of the underlying object and use methods of plane
    geometry to arrive at the solution. Here is one such example.
    Example: An ant lives on the surface of a regular tetrahedron with edges of length 3cm.
    It is currently at the midpoint of one of the edges and has to travel to the midpoint of the
    opposite edge where a grain is located (see figure). What is the length (in cm) of the
    shortest route to the destination assuming that the ant can only travel along the surface
    of the tetrahedron?
    Solution: The ant has several routes by which it can reach the grain. For instance, it can
    travel to the vertex C and move along edge CD.

    image1.png

    The idea behind finding the shortest route is to embed the surface of the tetrahedron on
    a plane. This is done by opening the tetrahedron along some edges and spreading it out.
    For example, the figure on the right is a planar representation containing the triangular
    faces ABC and ACD. Notice that ABCD is a rhombus of length 3cm and the segment
    joining ant and grain (which is the shortest route) is parallel to the base and thus of
    length 3cm as well.
    Now use the same idea to solve the problem below where the tetrahedron is replaced by
    a cube.
     
    sheetal2009, Feb 17, 2012 IP
  5. bigwriter

    bigwriter Active Member

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    #5
    (i) 18 cm.
    (ii) 3.5 cm
     
    bigwriter, Feb 19, 2012 IP
  6. explainstuff

    explainstuff Peon

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    #6
    Can the any not travel on the faces of the cube? Can it only go on the edges?
    If it can travel on the faces of the cube... then
    (1) The shortest distance to reach the grain is : 14 cms ... across the 2 faces. There are two distinct routes, one on the side that is visible in the picture and the other from the base of the cube and the side that is invisible to us.
     
    explainstuff, Feb 19, 2012 IP
  7. 5starpix

    5starpix Notable Member

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    #7
    (i) Since x is given as 2cm, it's pretty straight-forward. Just think of which path would be the quickest to get to the grain. (2+7+7+2) = 18cm

    (ii) You can use "x" as an unknown variable and solve for "x".

    28-2x = 14+2x
    28-14 = 2x+2x
    14 = 4x
    x = 3.5cm
     
    5starpix, Feb 19, 2012 IP