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
Solution
✅ 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 $l^3+b^3+h^3-3lbh=(l+b+h)\left(l^2+b^2+h^2-lb-bh-hl\right)$ $S-3V=22(196-144)$ $=22\times52$ $=1144$ $S-3V=1144\text{ cm}^3$