Question 49

Three cubes with integer edge lengths are given. It is known that the sum of their surface areas is 564 $$cm^{2}$$. Then the possible values of the sum of their volumes are

Solution

Let the sides of the three cubes be a, b, and c. It is known that the sum of their surface areas is 564 $$cm^{2}$$.

$$6\left(a^2+b^2+c^2\right)=564$$

$$\left(a^2+b^2+c^2\right)=94$$

Now, it is given that the sides of the cubes are integers, thus, (a,b,c) can be either (9,2,3) or (6,7,3)

Therefore, the sum of volumes can be either $$9^3+2^3+3^3=764\ cm^3$$ or it can be $$6^3+7^3+3^3=586\ cm^3$$

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