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Problem 194

AMC 10/12, early questions

Geometry Difficulty 3.7 Find the answer 16. tekmovanje v znanju matematike za dijake srednjih tehniških in strokovnih šol Državno tekmovanje · Slovenia

Problem:

Kocko z robom dolžine $a$ razrežemo na 8 malih skladnih kock. Površino ene take male kocke označimo s $P$ . Koliko je $P$ ?

(A) $\frac{a^{2}}{2}$
(B) $\frac{3 a^{2}}{4}$
(C) $\frac{a^{2}}{8}$
(D) $\frac{3 a^{2}}{2}$
(E) $a^{8}$

Your answer

Official solution

Solution:

Označimo z $a_{1}$ dolžino roba male kocke. Upoštevamo, da je prostornina male kocke $a_{1}^{3} = \frac{a^{3}}{8}$ in dobimo $a_{1} = \frac{a}{2}$ . Torej velja $P = 6 a_{1}^{2} = 6\left(\frac{a}{2}\right)^{2} = \frac{3 a^{2}}{2}$ .

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