# 1991 AJHSME Problems/Problem 15

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## Problem

All six sides of a rectangular solid were rectangles. A one-foot cube was cut out of the rectangular solid as shown. The total number of square feet in the surface of the new solid is how many more or less than that of the original solid?

$[asy] unitsize(24); draw((0,0)--(1,0)--(1,3)--(0,3)--cycle); draw((1,0)--(1+9*sqrt(3)/2,9/2)--(1+9*sqrt(3)/2,15/2)--(1+5*sqrt(3)/2,11/2)--(1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),4)--(1+2*sqrt(3),5)--(1,3)); draw((0,3)--(2*sqrt(3),5)--(1+2*sqrt(3),5)); draw((1+9*sqrt(3)/2,15/2)--(9*sqrt(3)/2,15/2)--(5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,5)); draw((1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),9/2)); draw((1+5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,11/2)); label("1'",(.5,0),S); label("3'",(1,1.5),E); label("9'",(1+9*sqrt(3)/4,9/4),S); label("1'",(1+9*sqrt(3)/4,17/4),S); label("1'",(1+5*sqrt(3)/2,5),E);label("1'",(1/2+5*sqrt(3)/2,11/2),S); [/asy]$

$\text{(A)}\ 2\text{ less} \qquad \text{(B)}\ 1\text{ less} \qquad \text{(C)}\ \text{the same} \qquad \text{(D)}\ 1\text{ more} \qquad \text{(E)}\ 2\text{ more}$

## Solution

Initially, that one-foot cube contributed 3 faces to the surface area of the whole solid. When it was removed, these 3 faces are removed, but there are 3 new faces where the cube was carved out, so the net change is $0 \rightarrow \boxed{\text{C}}$.