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
![=(31\times 17 +31\times -2x -2x\times 17 -2x\times -2x)x](https://tex.z-dn.net/?f=%3D%2831%5Ctimes%2017%20%2B31%5Ctimes%20-2x%20-2x%5Ctimes%2017%20-2x%5Ctimes%20-2x%29x)
![=(527 -62x-34x+4x^2)x](https://tex.z-dn.net/?f=%3D%28527%20-62x-34x%2B4x%5E2%29x)
![\implies V=4x^3-96x^2 +527x](https://tex.z-dn.net/?f=%5Cimplies%20V%3D4x%5E3-96x%5E2%20%2B527x)
Differentiating with respect to x,
![\frac{dV}{dx}=12x^2-192x+527](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdx%7D%3D12x%5E2-192x%2B527)
Again differentiating with respect to x,
![\frac{d^2V}{dx^2}=24x-192](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E2V%7D%7Bdx%5E2%7D%3D24x-192)
For maxima or minima,
![\frac{dV}{dx}=0](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdx%7D%3D0)
![\implies 12x^2-192x+527=0](https://tex.z-dn.net/?f=%5Cimplies%2012x%5E2-192x%2B527%3D0)
By the quadratic formula,
![x=\frac{192 \pm \sqrt{192^2 -4\times 12\times 527}}{24}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B192%20%5Cpm%20%5Csqrt%7B192%5E2%20-4%5Ctimes%2012%5Ctimes%20527%7D%7D%7B24%7D)
![x\approx 8\pm 4.4814](https://tex.z-dn.net/?f=x%5Capprox%208%5Cpm%204.4814)
![\implies x\approx 12.48\text{ or }x\approx 3.52](https://tex.z-dn.net/?f=%5Cimplies%20x%5Capprox%2012.48%5Ctext%7B%20or%20%7Dx%5Capprox%203.52)
Since, at x = 12.48,
= Positive,
While at x = 3.52,
= 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