Question

A candy is from a cardboard that measures 37 by 20 inches Squares of qual size will be cut out of each comer . The sides then folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum volume?

Answer 1

To find the square size cut from each corner for maximum volume of a rectangular box made from a 37 by 20 inch cardboard, one must use calculus to optimize the volume function V(x) = x(37 - 2x)(20 - 2x), take its derivative, find the critical points, and confirm a maximum.

Step-by-step explanation:

To maximize the volume of a rectangular box formed by folding up the sides of a piece of cardboard measuring 37 by 20 inches, after cutting squares from each corner, we need to determine the optimal size of the square cutouts. Let's denote the side length of the squares cut from each corner as x. After cutting the squares, the new dimensions of the box will be (37 - 2x) by (20 - 2x) by x, since the height of the box will be x after folding the sides. The volume V of the box can be expressed as a function of x: V(x) = x(37 - 2x)(20 - 2x).

To find the maximum volume, we need to take the derivative of the volume function with respect to x and set it to zero to find the critical points: V'(x) = 0. After finding the value of x that maximizes the volume, we need to verify that it is indeed a maximum by either checking the sign of the second derivative (concavity) at that point or by comparing volumes at values around the critical point.

Since this is a calculus optimization problem, it requires knowledge of differentiation and possibly also second derivatives or critical point analysis.