Choose the one alternative that best completes the statement or answers the question.
-From a 48-inch by 48 -inch piece of metal, squares are cut out of the four corners so that the sides can then be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are cut out. Express the volume of the box as a function of $x$.
A) $V ( x ) = 2 x ^ { 3 } - 144 x ^ { 2 } + 48 x$
B) $V ( x ) = 2 x ^ { 3 } - 144 x ^ { 2 }$
C) $V ( x ) = 4 x ^ { 3 } - 192 x ^ { 2 } + 2,304 x$
D) $V ( x ) = 4 x ^ { 3 } - 192 x ^ { 2 }$

Question

Quizplus · Accepted Answer

After cutting out squares of side length $x$ from each corner, the dimensions of the box will be $(48-2x)$ by $(48-2x)$ by $x$, since the height of the box will be $x$ and the length and width will each be reduced by $2x$ (once from each side). Therefore, the volume $V(x)$ of the box as a function of $x$ is given by $V(x) = x(48-2x)^2$. Expanding this, $V(x) = x(2304 - 192x + 4x^2) = 4x^3 - 192x^2 + 2304x$.

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