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
Options:
(A) (i), (ii), (iii), (iv)
(B) (ii), (i), (iv), (iii)
(C) (iii), (ii), (i), (iv)
(D) (iv), (iii), (ii), (i)
Q. 6 – Q. 10 carry two marks each.
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2018 Q.05)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Mensuration (Volume Comparison of 3D Geometric Solids)
**2. Detailed Step-by-Step Analytical Derivation**:
- Calculate the volume for each of the four given 3D objects using $\pi pprox rac{22}{7}$:
1. **(i) Cuboid**:
$V_{ ext{cuboid}} = l imes b imes h = 10 imes 8 imes 6 = 480 ext{ cm}^3$
2. **(ii) Cube**:
$V_{ ext{cube}} = a^3 = 8^3 = 512 ext{ cm}^3$
3. **(iii) Cylinder**:
$V_{ ext{cylinder}} = \pi r^2 h = rac{22}{7} imes 7^2 imes 7 = rac{22}{7} imes 49 imes 7 = 22 imes 49 = 1078 ext{ cm}^3$
4. **(iv) Sphere**:
$V_{ ext{sphere}} = rac{4}{3} \pi r^3 = rac{4}{3} imes rac{22}{7} imes 7^3 = rac{4}{3} imes 22 imes 49 = rac{4312}{3} pprox 1437.33 ext{ cm}^3$
- Arranging in descending order (largest to smallest volume):
$V_{ ext{sphere}} (1437.33) > V_{ ext{cylinder}} (1078) > V_{ ext{cube}} (512) > V_{ ext{cuboid}} (480)$
$ herefore ext{(iv)} > ext{(iii)} > ext{(ii)} > ext{(i)}$
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: Shows ascending order instead of descending order.
- **Option (B) [INCORRECT]**: Incorrect order placement.
- **Option (C) [INCORRECT]**: Places cylinder above sphere, which is incorrect.
- **Option (D) [CORRECT]**: Perfectly matches the descending order: (iv), (iii), (ii), (i).
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **D**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).
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