A square piece of cardboard is y = 14 inches long on each side. From each corner a square piece x inches on a side is cut out as shown. The flaps are then turned up to form an open box. Find a polynomial that represents the volume of the box.
A) $196 x - 56 x ^ { 2 } + x ^ { 3 }$
B) $196 x - 56 x ^ { 2 } + 4 x ^ { 3 }$
C) $196 x - 14 x ^ { 2 } + 4 x ^ { 3 }$
D) $196 - 56 x + 4 x ^ { 2 }$
E) $196 x - 56 x ^ { 2 } - 4 x ^ { 3 }$

Question

A square piece of cardboard is y = 14 inches long on each side. From each corner a square piece x inches on a side is cut out as shown. The flaps are then turned up to form an open box. Find a polynomial that represents the volume of the box. 
A) $196 x - 56 x ^ { 2 } + x ^ { 3 }$ 
B) $196 x - 56 x ^ { 2 } + 4 x ^ { 3 }$ 
C) $196 x - 14 x ^ { 2 } + 4 x ^ { 3 }$ 
D) $196 - 56 x + 4 x ^ { 2 }$ 
E) $196 x - 56 x ^ { 2 } - 4 x ^ { 3 }$

Quizplus · Accepted Answer

The volume of an open box is given by the formula $V=lwh$, where $l$, $w$, and $h$ are the length, width, and height respectively. In this case, the length and width of the cardboard are $14-2x$ (since $x$ is cut from each of the four corners) and the height of the box is $x$. Thus, we have: 
$V=(14-2x)(14-2x)(x)$
Simplifying: 
$V=4x(49-14x+x^2)$
$V=4x(x^2-14x+49)$
$V=4x^3-56x^2+196x$
which is equivalent to choice B.

A Square Piece of Cardboard Is Y = 14 Inches 196x−56x2+x3196 x - 56 x ^ { 2 } + x ^ { 3 }196x−56x2+x3

A Square Piece of Cardboard Is Y = 14 Inches $196 x - 56 x ^ { 2 } + x ^ { 3 }$