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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- Nov 6th 2008, 04:02 AMmagentaritaEdge 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?

- Nov 6th 2008, 04:55 AMHallsofIvy
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, $\displaystyle x^3$.

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: $\displaystyle (x+ 6)(x+ 12)(x- 4)= 2x^3$

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

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

Now, cubics, in general, are**hard**but we notice that the coefficient of $\displaystyle x^3$ 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 $\displaystyle 6^3- 14(6^2)+ 288= 0$. x= 6 is a solution. From that it is not to difficult to find that $\displaystyle x^3- 14x^2+ 288= (x- 6)(x^2- 8x+ 48)$ 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 $\displaystyle 6^3$= 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. - Nov 6th 2008, 03:43 PMmagentaritaok....