Geometry

Solids

Hard

The sum of length, breadth and height of a cuboid is 22 cm and length of its body diagonal is 14 cm. if S is the sum of cubes of dimensions of the cuboid and V is the volume, the (S−3V) will be equal to

572 cm³

728 cm³

984 cm³

1144 cm³

✅ Correct Option: 4

$l+b+h=22$

Using the formula for the body diagonal of a cuboid, $\sqrt{l^2+b^2+h^2}$ :

$\sqrt{l^2+b^2+h^2}=14$

$\Rightarrow l^2+b^2+h^2=196$

Using the identity $(l+b+h)^2=l^2+b^2+h^2+2(lb+bh+hl)$ :

$484=196+2(lb+bh+hl)$

$\Rightarrow 2(lb+bh+hl)=288$

$\Rightarrow lb+bh+hl=144$

Here $S=l^3+b^3+h^3$ and $V=lbh$ , so $S-3V$ is exactly the left side of the identity

$l^3+b^3+h^3-3lbh=(l+b+h)\left(l^2+b^2+h^2-lb-bh-hl\right)$

$S-3V=22(196-144)$

$=22\times52$

$=1144$

$S-3V=1144\text{ cm}^3$

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