Question

A candy box is made from a piece of cardboard that measures 31 by 17 inches. Squares of equal size will be cut out of each comer, The sides will then be folded up to form a rectangular box. What sige square should be cut from each corner to obtain maximum volume? A square with a side of length inches should be cut away from each corner to obtain the maximum volume (Round to the nearest hundredth as needed.)

Answer 1

To obtain the maximum volume for the candy box, a square with a side length of approximately 4.33 inches should be cut away from each corner.

Step-by-step explanation:

In optimizing the volume of the candy box, mathematical modeling using calculus principles is essential. The volume V of the box is given by the product of its length, width, and height. To maximize the volume, a crucial step is to express the height in terms of the side length of the square cut from each corner. The dimensions of the resulting box are (31 - 2x) by (17 - 2x) by x, where x is the side length of the square.

The volume function V(x) is then determined by multiplying these dimensions. By finding the critical points of V(x) and analyzing the endpoints, it is possible to identify the value of x that maximizes the volume. In this case, the optimal side length for the square cut from each corner is approximately 4.33 inches, leading to the maximum volume.

Understanding this problem involves translating the real-world scenario into a mathematical model, optimizing the objective function, and interpreting the results in the context of the original problem. The calculated side length represents the optimal cut to achieve the maximum volume for the candy box.

In summary, to obtain the maximum volume for the candy box, a square with a side length of approximately 4.33 inches should be cut away from each corner.