<p>A cubical cake with edge length <img alt="$2$" height="12" src="https://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png" width="8" /> inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where <img alt="$M$" height="12" src="https://latex.artofproblemsolving.com/5/d/1/5d1e4485dc90c450e8c76826516c1b2ccb8fce16.png" width="19" /> is the midpoint of a top edge. The piece whose top is triangle <img alt="$B$" height="12" src="https://latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" width="14" /> contains <img alt="$c$" height="8" src="https://latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" width="8" /> cubic inches of cake and <img alt="$s$" height="8" src="https://latex.artofproblemsolving.com/f/3/7/f37bba504894945c07a32f5496d74299a37aa51c.png" width="8" /> square inches of icing. What is <img alt="$c+s$" height="12" src="https://latex.artofproblemsolving.com/7/a/2/7a2d6387b6c5a4bb8dcfe21d0ebac1fb992d9198.png" width="38" />?</p> <p><img alt="[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); draw((1,1)--(-1,0)); pair P=foot((1,-1),(1,1),(-1,0)); draw((1,-1)--P); draw(rightanglemark((-1,0),P,(1,-1),4)); label("$M$",(-1,0),W); label("$C$",(-0.1,-0.3)); label("$A$",(-0.4,0.7)); label("$B$",(0.7,0.4)); [/asy]" height="98" src="https://latex.artofproblemsolving.com/b/9/a/b9a0b0d2b46c6d8034bbea244b0985f0df2a098d.png" width="118" /></p>

Question No: 278

A cubical cake with edge length $2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $M$ is the midpoint of a top edge. The piece whose top is triangle $B$ contains $c$ cubic inches of cake and $s$ square inches of icing. What is $c+s$ ?

$[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); draw((1,1)--(-1,0)); pair P=foot((1,-1),(1,1),(-1,0)); draw((1,-1)--P); draw(rightanglemark((-1,0),P,(1,-1),4)); label("$M$",(-1,0),W); label("$C$",(-0.1,-0.3)); label("$A$",(-0.4,0.7)); label("$B$",(0.7,0.4)); [/asy]$

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Tags: Algebra Geometry Probability

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