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Step-by-Step Solution
Step 1: Define the dimensions of the box
We start with a rectangular sheet of dimensions $a \times b$. We cut out squares of side $x$ from each of the four corners. The resulting box, when folded, will have:
Height $= x$
Length $= (a - 2x)$
Breadth $= (b - 2x)$
Step 2: Write the volume function
The volume $V$ of the box can be written as:
$V(x) = \text{(length)} \times \text{(breadth)} \times \text{(height)} = (a - 2x)(b - 2x)\,x.$
Step 3: Expand the volume function
Expand the expression for $V(x)$:
$V(x) = (a - 2x)(b - 2x) \, x \;=\; (ab - 2bx - 2ax + 4x^2)\,x.$
So,
$V(x) = abx - 2b x^2 - 2a x^2 + 4x^3 \;=\; 4x^3 - 2(a+b)x^2 + ab\,x.$
Step 4: Differentiate $V(x)$ with respect to $x$
To find the critical points, we compute $V'(x)$ and set it equal to zero:
$V'(x) = \frac{d}{dx} \bigl[4x^3 - 2(a+b)x^2 + ab\,x \bigr].$
Hence,
$V'(x) = 12x^2 - 4(a+b)x + ab.$
Step 5: Solve for critical points
Set $V'(x) = 0$:
$12x^2 - 4(a+b)x + ab = 0.$
Divide throughout by 4 to simplify:
$3x^2 - (a+b)x + \frac{ab}{4} = 0.$
Alternatively, solving the original form is also fine. The quadratic formula gives the roots as:
$x = \frac{4(a + b) \pm \sqrt{16(a + b)^2 - 48ab}}{2 \cdot 12}
\;=\; \frac{(a + b) \pm \sqrt{(a+b)^2 - 3ab}}{6}.$
Observe that $(a+b)^2 - 3ab = a^2 + b^2 + 2ab - 3ab = a^2 + b^2 - ab.$ So,
$x = \frac{(a + b) \pm \sqrt{a^2 + b^2 - ab}}{6}.$
Step 6: Identify the maximum volume root
We have two possible critical points:
$x_1 = \frac{(a + b) + \sqrt{a^2 + b^2 - ab}}{6}, \quad
x_2 = \frac{(a + b) - \sqrt{a^2 + b^2 - ab}}{6}.$
To ensure the box can be formed, $x$ must be less than $\frac{a}{2}$ and $\frac{b}{2}$ (since we cut squares of side $x$ from each corner). Among these two roots, the physically valid one that yields the maximum volume is:
$\displaystyle x = \frac{a + b - \sqrt{a^2 + b^2 - ab}}{6}.$
Step 7: Final Answer
Hence, if the box has maximum volume, the side of the square to be cut from each corner is:
$\displaystyle \boxed{ x = \frac{a + b - \sqrt{\,a^2 + b^2 - ab\,}}{6}. }$
Below are the reference pictures from the original reasoning steps:
