A rectangular box has a volume of $4320$ cubic inches and a surface area of $1704$ square inches. The sum of the lengths of its $12$ edges is $208$ inches. What would be the volume of the box, in cubic inches, if its length, width and height were each increased by one inch? plz help

To find the volume of the box if its length, width, and height are each increased by one inch, we need to solve the problem step by step.

Let's start by assigning variables to the dimensions of the current box. Let's say the length is $L$ inches, the width is $W$ inches, and the height is $H$ inches.

We know that the volume of a rectangular box is given by the formula:

Volume = Length × Width × Height

In this case, the volume is given as $4320$ cubic inches, so we have:

$4320 = L \times W \times H$

We also know that the surface area is given by the formula:

Surface Area = $2(LW + LH + WH)$

In this case, the surface area is given as $1704$ square inches, so we have:

$1704 = 2(LW + LH + WH)$

Furthermore, we are given that the sum of the lengths of the $12$ edges is $208$ inches. Each edge accounts for $2$ length measurements, so:

$208 = 2(2L + 2W + 2H)$

Now we have a system of three equations with three variables:

$4320 = L \times W \times H$

$1704 = 2(LW + LH + WH)$

$208 = 2(2L + 2W + 2H)$

To find the volume when each dimension is increased by one inch, we need to find the new dimensions of the box. We can solve the above system of equations to determine the current dimensions of the box.

Considering the equations, we can start by simplifying the third equation:

$208 = 8L + 8W + 8H$

Dividing both sides by $8$, we get:

$26 = L + W + H$

Next, let's rearrange the second equation:

$852 = LW + LH + WH$

Substituting $26$ for $L + W + H$, we have:

$852 = (L+W+H)(L+W) + WH$

Expanding and simplifying the equation, we get:

$852 = L^2 + W^2 + H^2 + 2(LW + LH + WH)$

Substituting $1704$ for $2(LW + LH + WH)$, we have:

$852 = L^2 + W^2 + H^2 + 2(1704)$

Simplifying further, we get:

$852 = L^2 + W^2 + H^2 + 3408$

Rearranging the equation, we have:

$L^2 + W^2 + H^2 = 852 - 3408$

$L^2 + W^2 + H^2 = -2556$

By substituting the value of $-2556$ into the equation, we find that the sum of the squares of the dimensions of the box is negative, which is not possible since lengths, widths, and heights are positive quantities. Hence, this system of equations has no real solutions, and we cannot determine the dimensions of the original box based on the given information.

Therefore, we cannot find the volume of the box if its length, width, and height are each increased by one inch.

look back to your previous post of this same problem