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?
Question
Answer
\(\left. \begin{array} { l } { \text { (a) } V ( x ) = x ( 24 - 2 x ) ^ { 2 } } \\ { \text { (b) } 972 \text { in } { } ^ { 3 } } \\ { \text { (c) } 160 \text { in } . ^ { 3 } } \\ { \text { (d) } V \text { is largest when } x = 4 } \end{array} \right.\)
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