GATE 2018 IN GA Q.5: solution and answer

GATE 2018 · INGA Q.5 · 1 mark · MCQ

Arrange the following three-dimensional objects in the descending order of their volumes:

(i) A cuboid with dimensions 10 cm, 8 cm and 6 cm

(ii) A cube of side 8 cm

(iii) A cylinder with base radius 7 cm and height 7 cm

(iv) A sphere of radius 7 cm

(A)
(i), (ii), (iii), (iv)
(B)
(ii), (i), (iv), (iii)
(C)
(iii), (ii), (i), (iv)
(D)
(iv), (iii), (ii), (i)

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Answer and solution

ANSWER

(D) (iv), (iii), (ii), (i): sphere (about 1437 cm^3) > cylinder (about 1078 cm^3) > cube (512 cm^3) > cuboid (480 cm^3).

Official answer: DSource: Official answer key, GATE 2018 IN (IIT Guwahati)

Cuboid, volume $= l \times b \times h$ : $V_{1} = 10 \times 8 \times 6 = 480 cm^{3}$ .
Cube, volume $= a^{3}$ : $V_{2} = 8^{3} = 512 cm^{3}$ .
Cylinder, volume $= π r^{2} h$ : $V_{3} = π \times 7^{2} \times 7 = 343 π \approx \frac{22}{7} \times 343 = 22 \times 49 = 1078 cm^{3}$ .
Sphere, volume $= \frac{4}{3} π r^{3}$ : $V_{4} = \frac{4}{3} π \times 7^{3} = \frac{4}{3} \times 343 π \approx \frac{4}{3} \times 1078 \approx 1437.3 cm^{3}$ .
Shortcut for (iii) and (iv): the cylinder has $h = r$ , so $V_{3} = π r^{3}$ , and $V_{4} = \frac{4}{3} π r^{3}$ is larger than it by a factor of $\frac{4}{3}$ . Since $π r^{3} = 343 π > 3 \times 343 = 1029 > 512$ , both are larger than the cube and the cuboid, whatever approximation of $π$ is used.
Descending order: $V_{4} > V_{3} > V_{2} > V_{1}$ , i.e. $1437.3 > 1078 > 512 > 480$ , which is (iv), (iii), (ii), (i). This is option (D).

KEY IDEA

Work out each volume with its standard formula: cuboid $l bh$ , cube $a^{3}$ , cylinder $π r^{2} h$ , sphere $\frac{4}{3} π r^{3}$ . The cylinder has $h = r = 7$ , so its volume is $π r^{3}$ and the sphere of the same radius is exactly $\frac{4}{3}$ times it. Both are more than $3 \times 343 = 1029 cm^{3}$ , far above the cube (512) and the cuboid (480), so the only close comparison is cube against cuboid.

WATCH OUT FOR

Guessing that the cuboid is bigger than the cube because its longest side (10 cm) is longer than the cube's. In fact $10 \times 8 \times 6 = 480 < 512 = 8^{3}$ . Another slip is using $π r^{2}$ or $\frac{4}{3} π r^{2}$ for the sphere and ending up with the sphere smaller than the cylinder. Also, the question asks for descending order, so reading it as ascending picks the reverse of the right sequence.

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GATE 2018 IN GA Q.5: solution and answer

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