Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 3131 in. by 1717 in. by cutting congruent squares from the corners and folding up the sides. then find the volume.

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

Differentiating with respect to x,

Again differentiating with respect to x,

For maxima or minima,

By the quadratic formula,

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