GATE 2022 IN GA Q.2: solution and answer

GATE 2022 · INGA Q.2 · 1 mark · MCQ

A sphere of radius $r$ cm is packed in a box of cubical shape.

What should be the minimum volume (in $cm^{3}$ ) of the box that can enclose the sphere?

(A)
$\frac{r ^{3}}{8}$
(B)
$r^{3}$
(C)
$2 r^{3}$
(D)
$8 r^{3}$

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Answer and solution

ANSWER

The smallest cube that can enclose the sphere has a side equal to the sphere's diameter, $2 r$ , so its volume is $(2 r)^{3} = 8 r^{3} cm^{3}$ . The answer is option (D).

Official answer: DSource: Official answer key, GATE 2022 IN (IIT Kharagpur)

The smallest cube that can hold a sphere is the one the sphere just fits inside. The sphere touches all six faces, so the cube's side equals the sphere's diameter.
Diameter of the sphere: $d = 2 r cm$ , so the cube's side is $a = 2 r cm$ .
Volume of a cube is its side cubed: $V = a^{3} = (2 r)^{3} = 8 r^{3} cm^{3}$
Check that nothing smaller works. Measured perpendicular to any pair of opposite faces, the sphere is $2 r$ wide, so a cube with side less than $2 r$ cannot hold it. The other options give sides of $\frac{r}{2}$ for (A), $r$ for (B) and $32 r \approx 1.26 r$ for (C). All three are less than $2 r$ , so they are ruled out.
Minimum volume $= 8 r^{3} cm^{3}$ , which is option (D).

KEY IDEA

A sphere inscribed in a cube touches all six faces, so the cube's side equals the sphere's diameter ( $2 r$ ), not its radius. Cube that side to get the volume.

WATCH OUT FOR

Taking the cube's side as the radius $r$ and choosing $r^{3}$ (B). Another slip is writing $2 r^{3}$ (C) instead of $(2 r)^{3} = 8 r^{3}$ , which cubes only the $r$ and not the factor 2.

From the GATE 2022 IN paper. Every question of this paper · Download the full paper · All 22 Geometry questions, with notes

GATE 2022 IN GA Q.2: solution and answer

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