An open box is constructed by starting with a rectangular sheet of metal $10$ in. by $14$ in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:
$\textbf{(A)}\ 140x - 48x^2 + 4x^3 \qquad  \textbf{(B)}\ 140x + 48x^2 + 4x^3\qquad \\ \textbf{(C)}\ 140x+24x^2+x^3\qquad \textbf{(D)}\ 140x-24x^2+x^3\qquad \textbf{(E)}\ \text{none of these}$
The resulting metal piece looks something like this where the white parts are squares of length $x$:
[asy] fill((0,4)--(4,4)--(4,0)--(6,0)--(6,4)--(10,4)--(10,10)--(6,10)--(6,14)--(4,14)--(4,10)--(0,10)--cycle,grey); draw((0,0)--(14-4,0)--(10,14)--(0,14)--cycle); draw((0,0)--(0,4)--(4,4)--(4,0)--cycle); draw((10-4,0)--(10,0)--(10,4)--(6,4)--cycle); draw((0,14)--(4,14)--(4,10)--(0,10)--cycle); draw((6,14)--(6,10)--(10,10)--(10,14)--cycle); [/asy]
From here, try to visualize the rectangular prism coming together and realize the height is $x$, the length is $14-2x$, and the width is $10-2x$. Therefore, the volume is $x(14-2x)(10-2x)=x(4x^2-48x+40)= \boxed{\textbf{(A) } 140x - 48x^2 + 4x^3}$.



[1957 AHSME](https://artofproblemsolving.com/wiki/index.php/1957_AHSME) ([Problems](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems) • [Answer Key](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Answer_Key) • [Resources](https://artofproblemsolving.comhttp://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=44&year=1957))


Preceded by[Problem 5](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_5)

Followed by[Problem 7](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_7)


[1](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_1) • [2](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_2) • [3](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_3) • [4](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_4) • [5](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_5) • 6 • [7](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_7) • [8](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_8) • [9](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_9) • [10](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_10) • [11](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_11) • [12](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_12) • [13](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_13) • [14](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_14) • [15](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_15) • [16](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_16) • [17](https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_17&action=edit&redlink=1) • [18](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_18) • [19](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_19) • [20](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_20) • [21](https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_21&action=edit&redlink=1) • [22](https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_22&action=edit&redlink=1) • [23](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_23) • [24](https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_24&action=edit&redlink=1) • [25](https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_25&action=edit&redlink=1) • [26](https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_26&action=edit&redlink=1) • [27](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_27) • [28](https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_28&action=edit&redlink=1) • [29](https://artofproblemsolving.com/wiki/index.php?title=1957_AHSME_Problems/Problem_29&action=edit&redlink=1) • [30](https://artofproblemsolving.com/wiki/index.php/1957_AHSME_Problems/Problem_30)


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