Richard has a 9-inch by 13.5-inch sheet of cardboard. he cuts squares from 4 corners to create an open container with the largest possible volume.
what is the greatest volume possible? express your answer as a decimal to the nearest cubic inch.

96.256 in³

Step-by-step explanation:

Given,

The dimension of the sheet = 9-inch by 13.5-inch,

Suppose x be the side of the square which is cut from each corner,

So, the dimension of the resultant open container,

(9-2x) inch × (13.5 - 2x) inch × x inch

Thus, the volume of the box,

V(x) = (9-2x) × (13.5 - 2x) × x

Differentiating with respect to x,

Again differentiating,

For maxima or minima,

Using the quadratic formula,

For x = 1.766, V''(x) = negative,

Thus, V(x) is maximum at x = 1.766,

For x =5.734, V''(x) = positive,

Thus, V(x) is minimum at x = 5.734,

Hence, the maximum possible volume, V(1.766) = (9-2(1.766)) × (13.5 - 2(1.766)) × 1.766

≈ 96.256 in³