62. A Box with an open top will be constructed of an 8 by 12 inch piece of cardboard by cutting out equal squares of side \( x \) at each corner and folding up. What is the domain that gives the max volume? a. \( 0

Ask by Franklin Hobbs. in the United States

Jan 23,2025

To maximize the volume of the box formed by cutting squares of side \( x \) from each corner, we need to consider the constraints imposed by the dimensions of the cardboard. Since the original dimensions are 8 inches in one direction and 12 inches in the other, cutting squares of side \( x \) will reduce each dimension accordingly. The largest possible \( x \) is limited by half of the smallest dimension, which is 4 inches. Thus, you can safely say that the domain for \( x \) must be between 0 and 4 inches, ensuring that the cuts don’t exceed the width or length of the cardboard. So the correct domain is \( 0 < x < 4 \), making option (a) the answer that ultimately maximizes the volume. To visualize why the upper limit is 4 inches, think about it this way: if you tried to cut squares larger than 4 inches, you wouldn't have any material left to fold up into a box! It would be like trying to create a sandwich with bread that's two sizes too small – it's just not going to happen!