Math Problem Statement
A box is formed by cutting squares from the four corners of a sheet of paper and folding up the sides.
A green sheet of paper is shown before and after a box is formed. Before the box is formed, the sheet has the square cutouts from each corner marked in gold. An arrow is drawn to the box formed by folding up the sides, and the gold square cutouts remain to help represent the dimensions of the box.
Suppose the paper is 9"-wide by 12"-long.
Estimate the maximum volume for this box. (Hint: Use your graphing calculator.)
Correct cubic inches
What cutout length produces the maximum volume?
Incorrect inches
Suppose we instead create the box from a 5"-wide by 7"-long sheet of paper.
Estimate the maximum volume for this box. (Hint: Use your graphing calculator.)
Correct cubic inches
What cutout length produces the maximum volume?
Incorrect inches
Solution
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Math Problem Analysis
Mathematical Concepts
Algebra
Volume Optimization
Quadratic Functions
Formulas
Volume of a rectangular box: V = (length - 2x)(width - 2x)(x)
Quadratic equation for volume maximization
Theorems
Critical points theorem
First derivative test for optimization
Suitable Grade Level
Grades 10-12
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