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Astronaut puzzle - help!

Discussion in 'Mathematics' started by yorksliz, Sep 3, 2011.





  1. <font size="3" face="Times New Roman">

    </font>I have a set of investigations which I was
    given during teacher training and have finally got round to having a proper look
    at. This is one I would like to use this
    week as we spoke on Friday about the recent rocket record breaking attempt,
    however, I can&rsquo;t figure it out! I even
    have a &lsquo;solution&rsquo; which I have put below the question but I am obviously still
    in holiday mode and missing something crucial. I can get to 63.2 miles but no
    further. Can anyone help?</font>
    <font size="3" face="Times New Roman">

    </font>
    <font size="3" face="Times New Roman">

    </font>"A spacecraft has crash landed on the Moon
    67 miles from its intended target, the Moon Base at Armstrong Crater.</font>



    <font size="3" face="Times New Roman">

    </font>The astronaut must leave his craft and make
    his way unaided to the Moon Base.</font>



    <font size="3" face="Times New Roman">

    </font>The spacecraft has on board, 12 cylinders
    full of oxygen. </font>



    <font size="3" face="Times New Roman">

    </font>These cylinders could be attached to the
    astronaut&rsquo;s back and carried by him. Each cylinder can take the astronaut a
    distance of 20 miles and no more. </font>



    <font size="3" face="Times New Roman">

    </font> Unfortunately the astronaut can only carry two
    cylinders at a time. There is nothing to
    stop him leaving a cylinder a distance from the craft and returning for others.</font>
    <font size="3" face="Times New Roman">

    </font>
    <font size="3" face="Times New Roman">

    </font>Is it possible for the astronaut to reach
    the Moon Base ?</font>



    <font size="3" face="Times New Roman">

    </font>



    <font size="3" face="Times New Roman">

    </font>Need to get two cylinders to the
    point 40 miles from the base. This is achievable if you can get 3 cylinders to
    the point 46.7 miles from the base. You can do this if you can get 6 cylinders
    to the point 56.7 miles from the base and this can be done with the 12
    cylinders you started with.
    </font>
     




  2. <font size="3" face="Times New Roman">

    </font>I have a set of investigations which I was
    given during teacher training and have finally got round to having a proper look
    at. This is one I would like to use this
    week as we spoke on Friday about the recent rocket record breaking attempt,
    however, I can&rsquo;t figure it out! I even
    have a &lsquo;solution&rsquo; which I have put below the question but I am obviously still
    in holiday mode and missing something crucial. I can get to 63.2 miles but no
    further. Can anyone help?</font>
    <font size="3" face="Times New Roman">

    </font>
    <font size="3" face="Times New Roman">

    </font>"A spacecraft has crash landed on the Moon
    67 miles from its intended target, the Moon Base at Armstrong Crater.</font>



    <font size="3" face="Times New Roman">

    </font>The astronaut must leave his craft and make
    his way unaided to the Moon Base.</font>



    <font size="3" face="Times New Roman">

    </font>The spacecraft has on board, 12 cylinders
    full of oxygen. </font>



    <font size="3" face="Times New Roman">

    </font>These cylinders could be attached to the
    astronaut&rsquo;s back and carried by him. Each cylinder can take the astronaut a
    distance of 20 miles and no more. </font>



    <font size="3" face="Times New Roman">

    </font> Unfortunately the astronaut can only carry two
    cylinders at a time. There is nothing to
    stop him leaving a cylinder a distance from the craft and returning for others.</font>
    <font size="3" face="Times New Roman">

    </font>
    <font size="3" face="Times New Roman">

    </font>Is it possible for the astronaut to reach
    the Moon Base ?</font>



    <font size="3" face="Times New Roman">

    </font>



    <font size="3" face="Times New Roman">

    </font>Need to get two cylinders to the
    point 40 miles from the base. This is achievable if you can get 3 cylinders to
    the point 46.7 miles from the base. You can do this if you can get 6 cylinders
    to the point 56.7 miles from the base and this can be done with the 12
    cylinders you started with.
    </font>
     
  3. Use 6 cylinders to get the remaining 6 to the first 'drop-off' point. 6 cylinders can travel a total of 120 miles and will have to make 11 journeys between start and first drop-off point. Therefore the first drop off point can be 120/11 miles from the start.
    Then use 3 cylinders to get the remaining 3 to the next 'drop-off' point. 3 cylinders can travel 60 miles and will have to make 5 journeys between first and second drop-off point. Therefore the second drop off point can be 12 miles (60/5) from the first drop off point.
    Next use 1 cylinder to get the last 2 to the third drop off point. 1 cylinder can travel 20 miles and will have to make 3 journeys between the second and third drop-off point. Therefore, the third drop off point can be 20/3 miles from the start.
    Finally, the astronaut can put the 2 remaining cylinders on his back knowing that he can travel up to 40 miles...and hopefully 120/11 + 12 + 20/3 + 40 > 67.
    I hope this makes sense...and further more hope that it is correct, otherwise I will be returning to the drawing board!!!!

     
  4. Thanks mathmo; that makes perfect sense. I had initially got fixed on having to move
    the cylinders in 10 mile stages thus having half tanks left over. Then I looked at using one tank to move all
    the rest, then two&hellip;now you have explained the solution it seems so obvious!
    <font size="3" face="Times New Roman">

    </font>


    Out of interest &ndash; would you expect a good S2 set to be able
    to do this? (Top set, hard working, 32 pupils).
    How long do you think before the fastest got it?





    Thanks again [​IMG]
     
  5. I only teach up to KS3 so not sure really!! It's a great problem though!!
     
  6. Maths_Mike

    Maths_Mike New commenter

    excellent solution so please dont take this as criticism and I dont know the asnwer its a genuine question.
    It seems obvious to use 6 cylinders to move the remaining 6 but why?
    Why not use 4 cylinders to move 8 or any other combination - how did you know what that optimum was?
     
  7. I think this problem is out of this world
     
  8. Interesting question Maths_Mike - made me look at this further using your suggested 4 cylinders to move 8. Using same format as mathmo2000's excellent solution, would give:
    use 4 cylinders to move 8 to first drop off point. 15 journeys required so can travel 80 / 15 km.
    use 4 cylinders to move 4 to second drop off point. 7 journeys required so can travel 80 / 7 km.
    use 2 cylinders to move 2 to final drop off point. 3 journeys required so can travel 40 / 3 miles.
    Gives (80/15 + 80/7 + 40/3 + 20 + 20) >67
    Distance travelled here is slightly greater than first solution, but both would be acceptable.
    Original question just asked to travel more than 67km. An extension could be to ask for optimal solution/max possible distance can travel. Although I'm not sure what they answer would be! More thought needed.
     




  9.  
  10. <font face="Calibri">I have been struggling with this &ndash; the solution says that it is possible, but does not explain how to get the canisters &lsquo;56.7 miles from the base&rsquo;. Again, mathmo2000 shows that it should be possible, but does not show how. Thinking about mathmo2000&rsquo;s ideas helped to unstick me.</font><font face="Calibri">This is my solution.</font><font face="Calibri">First drop-off is 10 miles from the start. I now have 6 full canisters and one half canister &ndash; it is this canister that I use to get all the others as far as possible i.e. 10/11 miles.</font><font face="Calibri">Second drop-off is 10.9 miles from the start. I now have 6 full canisters. I use these to travel a further 10 miles.</font><font face="Calibri">Third drop-off is 20.9 miles from the start. I have 3 full canisters and one half canister. I use one and a half canisters to move 10 miles.</font><font face="Calibri">Fourth drop-off is 30.9 miles from the start. I have 2 full canisters, which will take me 40 miles, so I can travel 70.9 miles and make the base.</font><font face="Calibri">Blue.</font>
     

  11. First drop-off is 10 miles from the start, I use 5.5 canisters in order to get there. I now have 6 full canisters and one half canister &ndash; it is this canister that I use to get all the others as far as possible i.e. 10/11 miles.
    of these to travel a further 10 miles.
    Third drop-off is 20.9 miles from the start. I have 3 full canisters and one half canister. I use one and a half canisters to move 10 miles.
    Fourth drop-off is 30.9 miles from the start. I have 2 full canisters, which will take me 40 miles, so I can travel 70.9 miles and make the base.


     
  12. My solution is flawed as has been pointed out but subsequent solutions are making sense!!! Damn, I felt like a proper mathematician for a moment!!!
     
  13. This reminds me of the camel and banana problem:

    "A camel is sitting by a stack of 3000 bananas at edge of a 1000 mile wide desert. He is going to travel across the desert carrying as many bananas as he can to the other sides. He can carry 1000 bananas at a time, but he eats one banana every mile.

    What is the maximum number of bananas the camel can get across the desert? How does the camel do it? Explain how you arrived to your answer."

     

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