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Math Help - When each edge of a cube is decreased by 1 cm, its volume is decreased by 91 cm^3.

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    When each edge of a cube is decreased by 1 cm, its volume is decreased by 91 cm^3.

    Hello! I would like some help with this task.

    When each edge of a cube is decreased by 1 cm, its volume is decreased by 91 cm^3. Find the length of a side of the orginal cube.

    Thank you!
    Last edited by mr fantastic; February 1st 2011 at 11:31 AM. Reason: Re-titled.
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    V_1 = a^3 and V_2 = (a-1)^3

    V_2 = V_1 - 91 so V_1 - 91 = a^3 - (a-1)^3

    edit: a^3 will cancel but you'll need the binomial expansion to expand (a-1)^3
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    I do not understand this V1 - 91 = a^3 - (a-1)^3 can you please explain?
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    Quote Originally Posted by Anna55 View Post
    I do not understand this V1 - 91 = a^3 - (a-1)^3 can you please explain?
    Hi Anna55,

    The first volume is V_1=a^3

    The volume after side reduction is V_2=(a-1)^3

    The difference between the two volumes is a^3-(a-1)^3=91

    Expand and solve. Throw out the negative result.
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    Quote Originally Posted by Anna55 View Post
    I do not understand this V1 - 91 = a^3 - (a-1)^3 can you please explain?
    It means just what it says- cube a and a-1 and subtract. Do you know how to multiply a- 1 by itself? However, the "V1- 91" is incorrec- perhaps that is what is confusing you. It should be a^3- (a-1)^3= 91. Multiply that out and solve for a. As e^(i pu) said, the a^4 terms will cancel leaving a quadratic equation to solve.

    If you do not know how to multiply (a- 1)^3, get back to us.
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    Thank you very much HallsofIvy and masters , I understand everything now.
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    Quote Originally Posted by Anna55 View Post
    Hello! I would like some help with this task.

    When each edge of a cube is decreased by 1 cm, its volume is decreased by 91 cm^3. Find the length of a side of the orginal cube.

    Thank you!
    hi Anna,

    you can also calculate this with a quadratic equation.

    Suppose all the sides of the cube is x.
    Now, cut off a vertical slice of 1 unit deep (say from the right-hand side of the cube).

    The volume of the part cut off is x(x)(1)=x^2 cubic units.

    Next, cut off a piece from the front or back (say from the back).
    One side length is now (x-1) units long,
    therefore the volume of the piece cut off is x(x-1)(1)=x^2-x cubic units.

    Finally cut off a piece from the top.
    This time, 2 sides have length (x-1), so the volume of the piece cut off is

    (1)(x-1)^2=x^2-2x+1 cubic units.

    The total volume of the removed parts is x^2+x^2-x+x^2-2x+1=3x^2-3x+1

    Hence

    3x^2-3x+1=91\Rightarrow\ 3x^2-3x-90=0\Rightarrow\ x^2-x-30=0

    \Rightarrow\ (x-6)(x+5)=0\Rightarrow\ x=6\;cm


    You could of course try a shortcut, which works only if the side length is an integer.

    2^3-1^3=8-1=7

    3^3-2^3=27-8=19

    4^3-3^3=64-27=37

    5^3-4^3=125-64=61

    and so on.
    Last edited by Archie Meade; February 2nd 2011 at 01:49 AM.
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