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STD 11 - rectangular cartensian co-ordinates — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - rectangular cartensian co-ordinates4 Marks
Question
Consider a cuboid all of whose edges are integers and whose base is a square. Suppose the sum of all its edges is numerically equal to the sum of the areas of all its six faces. Then, the sum of all its edges is
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Answer

c
(c)

Given, a cuboid has all edges are integers and base is square.

Let the length, breadth and height of cuboid is $x, x, y$.

Sum of all edges of cuboid $=4 x+4 x+4 y$

Sum of area of all faces $=2 x^2+2 x y+2 x y$

Given,

Sum of all edges of cuboid = Sum of area of all faces

$\therefore \quad 4 x+4 x+4 y=2\left(x^2+x y+x y\right)$

$\Rightarrow \quad 4(2 x+y)=2\left(x^2+2 x y\right)$

$\Rightarrow \quad x^2+2 x y-4 x-2 y=0$

$\Rightarrow \quad x^2+2 x(y-2)-2 y=0$

$\Rightarrow \quad x=\frac{-2(y-2) \pm \sqrt{4(y-2)^2+4(2 y)}}{2}$

$\Rightarrow \quad x=y-2 \pm \sqrt{y^2-2 y+4}$

$x \text { is integer, when } y=2$

$\therefore \quad \quad y=2 x=2$

Hence, sum of edges $=8 x+4 y=16+8=24$

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