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Math Help - Edge of a Cube

  1. #1
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    Edge of a Cube

    What is the length of the edge of a cube if its volume could be doubled by an increase of 6 centimeters in one edge, an increase of of 12 centimeters in a second edge and a decrease of 4 centimeters in the third edge?
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    Quote Originally Posted by magentarita View Post
    What is the length of the edge of a cube if its volume could be doubled by an increase of 6 centimeters in one edge, an increase of of 12 centimeters in a second edge and a decrease of 4 centimeters in the third edge?
    Start by writing equations! Since the problem asks "what is the length of the edge", write "Let x be the length of a side of the cube, in centimeters".
    Then the volume of that cube is, of course, x^3.

    Increasing one side by 6 cm changes from x to x+ 6.

    Increasing a second side by 12 cm changes from x to x+ 12.

    Decreasing the third side by 4 cm changes from x to x- 4.

    The volume of this rectangular solid is (x+6)(x+ 12)(x- 4) and we are told that is double the original volume: (x+ 6)(x+ 12)(x- 4)= 2x^3

    (x+ 6)(x+ 12)= x^2+ 18x+ 72 and multiplying that by x- 4 gives x^3+ 14x^2- 288

    So we have x^3+ 14x^2- 288= 2x^3 which is the same as x^3- 14x^2- 288= 0. We need to solve that equation.

    Now, cubics, in general, are hard but we notice that the coefficient of x^3 is 1 so any rational number solution, if there is one, must be an integer factor of 288. Crossing our fingers and hoping there is such a solution, we try x= 1, 2, 3, 4, 6, etc. and fortunately find that 6^3- 14(6^2)+ 288= 0. x= 6 is a solution. From that it is not to difficult to find that x^3- 14x^2+ 288= (x- 6)(x^2- 8x+ 48) and that quadratic has no real number solutions. The only solution is x= 6.

    Now be sure to write the answer clearly:
    The cube must have edges of length 6 cm.

    and check:
    If the original cube has edge length 6 cm. then its volume is 6^3= 216 cubic centimeters.

    Increasing the one length by 6 cm, another by 12 cm, and reducing the third by 4 cm would give a rectangular solid (6+6) by (6+ 12) by (6- 2)= 12 by 18 by 2 and that has volume (12)(18)(2)= 432 cubic centimeters = 2(216) cubic centimeters, as we said.
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  3. #3
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    ok....

    Quote Originally Posted by HallsofIvy View Post
    Start by writing equations! Since the problem asks "what is the length of the edge", write "Let x be the length of a side of the cube, in centimeters".
    Then the volume of that cube is, of course, x^3.

    Increasing one side by 6 cm changes from x to x+ 6.

    Increasing a second side by 12 cm changes from x to x+ 12.

    Decreasing the third side by 4 cm changes from x to x- 4.

    The volume of this rectangular solid is (x+6)(x+ 12)(x- 4) and we are told that is double the original volume: (x+ 6)(x+ 12)(x- 4)= 2x^3

    (x+ 6)(x+ 12)= x^2+ 18x+ 72 and multiplying that by x- 4 gives x^3+ 14x^2- 288

    So we have x^3+ 14x^2- 288= 2x^3 which is the same as x^3- 14x^2- 288= 0. We need to solve that equation.

    Now, cubics, in general, are hard but we notice that the coefficient of x^3 is 1 so any rational number solution, if there is one, must be an integer factor of 288. Crossing our fingers and hoping there is such a solution, we try x= 1, 2, 3, 4, 6, etc. and fortunately find that 6^3- 14(6^2)+ 288= 0. x= 6 is a solution. From that it is not to difficult to find that x^3- 14x^2+ 288= (x- 6)(x^2- 8x+ 48) and that quadratic has no real number solutions. The only solution is x= 6.

    Now be sure to write the answer clearly:
    The cube must have edges of length 6 cm.

    and check:
    If the original cube has edge length 6 cm. then its volume is 6^3= 216 cubic centimeters.

    Increasing the one length by 6 cm, another by 12 cm, and reducing the third by 4 cm would give a rectangular solid (6+6) by (6+ 12) by (6- 2)= 12 by 18 by 2 and that has volume (12)(18)(2)= 432 cubic centimeters = 2(216) cubic centimeters, as we said.
    I thank you very much for taking time out to help me understand this question a little better than before.
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