Answer:
840.02 square inches ( approx )
Step-by-step explanation:
Suppose x represents the side of each square, cut from the corners of the sheet,
Since, the dimension of the sheet are,
31 in × 17 in,
Thus, the dimension of the rectangular box must are,
(31-2x) in × (17-2x) in × x in
Hence, the volume of the box would be,
V = (31-2x) × (17-2x) × x
[tex]=(31\times 17 +31\times -2x -2x\times 17 -2x\times -2x)x[/tex]
[tex]=(527 -62x-34x+4x^2)x[/tex]
[tex]\implies V=4x^3-96x^2 +527x[/tex]
Differentiating with respect to x,
[tex]\frac{dV}{dx}=12x^2-192x+527[/tex]
Again differentiating with respect to x,
[tex]\frac{d^2V}{dx^2}=24x-192[/tex]
For maxima or minima,
[tex]\frac{dV}{dx}=0[/tex]
[tex]\implies 12x^2-192x+527=0[/tex]
By the quadratic formula,
[tex]x=\frac{192 \pm \sqrt{192^2 -4\times 12\times 527}}{24}[/tex]
[tex]x\approx 8\pm 4.4814[/tex]
[tex]\implies x\approx 12.48\text{ or }x\approx 3.52[/tex]
Since, at x = 12.48, [tex]\frac{d^2V}{dx^2}[/tex] = Positive,
While at x = 3.52, [tex]\frac{d^2V}{dx^2}[/tex] = Negative,
Hence, for x = 3.52 the volume of the rectangle is maximum,
Therefore, the maximum volume would be,
V(3.5) = (31-7.04) × (17-7.04) × 3.52 = 840.018432 ≈ 840.02 square inches
