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Constructing an Open Box An open box with a square base is to be made from a square piece of cardboard $$24$$ inches on a side by cutting out a square from each corner and turning up the sides. See the figure. (a) Express the volume $$V$$ of the box as a function of the length $$x$$ of the side of the square cut from each corner. (b) What is the volume if a $$3$$ -inch square is cut out? (c) What is the volume if a $$10$$ -inch square is cut out? Graph $$V = V ( x )$$ . For what value of $$x$$ is $$V$$ largest?
$$\left. \begin{array} { l } { \text { (a) } V ( x ) = x ( 24 - 2 x ) ^ { 2 } } \\ { \text { (b) } 972 \text { in } { } ^ { 3 } } \\ { \text { (c) } 160 \text { in } . ^ { 3 } } \\ { \text { (d) } V \text { is largest when } x = 4 } \end{array} \right.$$