Math Problem Statement

A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.

Solution

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Math Problem Analysis

Mathematical Concepts

Optimization
Volume Calculation
Derivatives

Formulas

Volume of the box: V = (3 - 2x)^2 * x
Derivative: V'(x) = 9 - 24x + 12x^2
Quadratic equation: ax^2 + bx + c = 0

Theorems

Critical points in optimization
Quadratic formula

Suitable Grade Level

Grades 11-12