Question

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 $12$centimeters in a second edge, and

a decrease of $4$centimeters in the third edge?

Short answer

If the volume of the cube increases to double by increasing one side of the cube by$6$, increasing the second side of the cube by$12$and decreasing the third side of the cube is by $4$then the side of the cube might be $6$or $12$centimeter

Step-by-step solution

Step 1. Given data

one side of the cube increased by $6cm$

the second side of the cube increased by $12cm$

the third side of the cube is decreased by $4cm$

the volume of the cube increases to double by changing sides

Step 2. Formation of equation 

consider a side of the cube isx

then volume $V=x^3$

After changing sides

$$\begin{gathered} (x+6)(x+12)(x-4)=2V \\ (x+6)(x+12)(x-4)=2x^3 \\ {x}^3-4x^2+18x^2-72x+72x-288=0 \\ {x}^3-14x^2+288=0 \end{gathered}$$

The degree of the equation is three so the possible maximum number of real zeros is$3$

Step 3. possible zeros 

the constant term is$288$

the leading coefficient is $1$

so potential rational zeros are

$\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 8,\pm 9,\pm 12,\pm 16,\pm 18,\pm 24,\pm 32,\pm 36,\pm 48,\pm 72,\pm 96,\pm 144,\pm 288$

Step 4. testing potential rational zeros 

Test potential rational zeros $6$and factor the equation by synthetic division

$\frac{6\begin{array}{cccc}1& -14& 0& 288\\ & 6& -48& -288\end{array}}{\begin{array}{cccc}1& -8& -48& 0\end{array}}$

so$f\left(6\right)=0$and $x=6$is a solution of equation

the equation can be factorized as

$(x-6)(x^2-8x-48)=0$

Step 5. factorize the depressed equation

Factorized the equation$x^2-8x-48=0$

role="math" localid="1646133756000" $$\begin{gathered} x^2-8x-48=0 \\ {x}^2-12x+4x-48=0 \\ x(x-12)+4(x-12)=0 \\ (x+4)(x-12)=0 \end{gathered}$$

separate the factors

role="math" localid="1646133767210" $$\begin{gathered} x+4=0 \\ x=-4 \end{gathered}$$

but length cannot be negative

and role="math" localid="1646133804257" $$\begin{gathered} x-12=0 \\ x=12 \end{gathered}$$

so the length of the cube could be$6$or$12$centimeter