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Math Help - More Word Problems! Help!!

  1. #1
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    Unhappy More Word Problems! Help!!



    6. A 2-by-4-by-8 rectangular solid is painted red. It is cut into unit cubes and reassembled into a 4-by-4-by-4 cube. If the entire surface of this cube is red, how many painted unit-cube faces are hidden in the interior cube?


    7. If a year has 364 days, then the same calendar could be used every year by only changing the year. A "regular" year has 365 days and a leap year has 366 days. The year 2000 has a leap year and leap years occur every 4 years between the years 2000 and 2100. Claudia has calendar for 2009. What will be the next year that she can use this calendar by merely changing the year?






    8. A 6-question True-False test has True as the correct answer for at least 2/3 of the questions. How many different true/False answer patterns are possible on an answer key for this test?
    Last edited by Brent; November 21st 2008 at 05:58 PM.
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  2. #2
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    Hello, Brent!

    It's simple arithmetic . . .


    6. A 2-by-4-by-8 rectangular solid is painted red.
    It is cut into unit cubes and reassembled into a 4-by-4-by-4 cube.
    If the entire surface of this cube is red,
    how many painted unit-cube faces are hidden in the interior cube?
    Code:
                     8
              * - - - - - - *
             /             /|
          4 /             / | 2
           /             /  |
          * - - - - - - *   *
          |             |  /
        2 |             | / 4
          |             |/
          * - - - - - - *
                 8
    The total surface area is:

    . . \begin{array}{cccccc}\text{top/bottom:} & 2\times(8\times4) &=&64 \\<br />
\text{left/right:} & 2\times(4\times2) &=& 16 \\<br />
\text{front/back:} & 2\times(8\times2) &=&32 \\\end{array}\quad\Rightarrow\quad 112

    There are 112 red faces on the sixty-four unit cubes.



    Code:
              * - - - - *
             /         /|
           4/         / |
           /         /  | 4
          * - - - - *   |
          |         |   *
        4 |         |  /
          |         | /4
          |         |/
          * - - - - *
               4
    The total surface area of the cube is: .  6 \times (4\times4) \:=\:96

    There are 96 red faces on the outside of the cube.


    Therefore, there are: . 112 - 96 \:=\:\boxed{16} red faces hidden inside.

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