From one of the faces of a solid wooden cube of side 14 cm, maximum number of hemispheres of diameter 1.4 cm are scooped out. Find the total number of hemispheres that can be scooped out. Also, find the total surface area of the remaining solid.
Generated by claude-sonnet-4-6 · 2026-06-15 10:35 · grounding rag
Model Answer
Step 1: Number of hemispheres
Side of cube = 14 cm, diameter of hemisphere = 1.4 cm, so radius $r$ = 0.7 cm.
Number of hemispheres along one edge = $\dfrac{14}{1.4} = 10$
Total hemispheres on one face = $10 \times 10 = \mathbf{100}$
Step 2: Total Surface Area of remaining solid
TSA = (Surface area of cube) – (Area of 100 circular bases on one face) + (CSA of 100 hemispheres)
- TSA of cube $= 6 \times 14^2 = 1176 \text{ cm}^2$
- Area of 100 circular holes $= 100 \times \pi r^2 = 100 \times \dfrac{22}{7} \times (0.7)^2 = 100 \times 1.54 = 154 \text{ cm}^2$
- CSA of 100 hemispheres $= 100 \times 2\pi r^2 = 2 \times 154 = 308 \text{ cm}^2$
$$\text{TSA} = 1176 - 154 + 308 = \mathbf{1330 \text{ cm}^2}$$
Source: Surface Area of a Combination of Solids, Chapter 12
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Explanation
- The key idea is that scooping removes the flat circular area from the face but adds the curved surface of each hemisphere.
- Formula used: TSA of remaining solid = TSA of cube − (100 × πr²) + (100 × 2πr²)
- Examiners award marks for: correct count (100), correct formula setup, and accurate arithmetic. Show each step clearly.