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On each face of a cuboid, the sum of its perimeter and its area is written. Among the six numbers so written, there are three distinct numbers and they are $16,24$ and $31$. The volume of the cuboid lies between
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The correct answer is:
$28$ and $35$
$2(a+b)+a b=16 \quad\dots(1)$
$2(b+c)+b c=24 \quad\dots(2)$
$2(c+a)+c a=31 \quad\dots(3)$
From equation $(2)$, equation
$(3) \Rightarrow(a-b)(2+c)=7 \quad\dots(4)$
From equation $(2)$ and equation
$(4) \Rightarrow 4 a=2+5 b \quad\dots(5)$
Solve equation $(1)$ and $(5)$
$b=2 a=3, c=5$
Volume $=30$
$2(b+c)+b c=24 \quad\dots(2)$
$2(c+a)+c a=31 \quad\dots(3)$
From equation $(2)$, equation
$(3) \Rightarrow(a-b)(2+c)=7 \quad\dots(4)$
From equation $(2)$ and equation
$(4) \Rightarrow 4 a=2+5 b \quad\dots(5)$
Solve equation $(1)$ and $(5)$
$b=2 a=3, c=5$
Volume $=30$
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