CAT 2025 Slot 3 QA Question 22: The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm, and the sum of the lengths of all its edges is 144 cm. The volume, in cubic cm, of the sphere is

Question

The surface area of a closed rectangular box, which is inscribed in a sphere, is 846 sq cm, and the sum of the lengths of all its edges is 144 cm. The volume, in cubic cm, of the sphere is

Accepted Answer

Let the dimensions of the rectangular box be $l$, $b$ and $h$.
Surface area = $2(lb + bh + lh) = 846$
$\Rightarrow 2(lb + bh + lh) = 846$
Sum of all edges = $4(l + b + h) = 144$
$\Rightarrow l + b + h = 36$
The body diagonal of the cuboid is the diameter of the sphere.
$l^2 + b^2 + h^2 = (l + b + h)