Solution Found!
Folded boxes (a). Squares with sides of length x are cut
Chapter 7, Problem 26E(choose chapter or problem)
a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.
b. Suppose that in part (a) the original piece of cardboard is a square with sides of length \(\ell\). Find the volume of the largest box that can be formed in this way.
c. Suppose that in part (a) the original piece of cardboard is a rectangle with sides of length \(\ell\) and L. Holding \(\ell\) fixed, find the size of the corner squares x that maximizes the volume of the box as \(L \rightarrow \infty\). (Source: Mathematics Teacher, November 2002)
Questions & Answers
QUESTION:
a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.
b. Suppose that in part (a) the original piece of cardboard is a square with sides of length \(\ell\). Find the volume of the largest box that can be formed in this way.
c. Suppose that in part (a) the original piece of cardboard is a rectangle with sides of length \(\ell\) and L. Holding \(\ell\) fixed, find the size of the corner squares x that maximizes the volume of the box as \(L \rightarrow \infty\). (Source: Mathematics Teacher, November 2002)
ANSWER:Solution 26E Step 1: (a) onsider abcd is the rectangular sheet of paper such thatab andcd are the lengths and ac andbd are the widths. If a square with sides of length x is cut from each corner, then the new dimensions of the sheet measuring 3ft by 4ft will be in length and in width. Consider the following figure