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Mathematics · 2025 · 5 marks
CBSE 2025 · Region 4 · Set 1 · Q34
A wooden cubical die is formed by forming hemispherical depressions on each face of the cube such that face 1 has one depression, face 2 has two…
A wooden cubical die is formed by forming hemispherical depressions on each face of the cube such that face $\displaystyle 1$ has one depression, face $\displaystyle 2$ has two depressions and so on. The sum of number of hemispherical depressions on opposite faces is always 7. If the edge of the cubical die measures $\displaystyle 5$ cm and each hemispherical depression is of diameter $\displaystyle 1.4$ cm, find the total surface area of the die so formed.
Marking-scheme solution
Number of hemispherical depressions = $\displaystyle 21$
Total surface area of the die formed
= TSA of cube + CSA of $\displaystyle 21$ hemispheres - Base area of $\displaystyle 21$ hemispherical depressions
\[\begin{aligned}
& =6 \times 5^{2}+21 \times 2 \times \frac{22}{7} \times(0.7)^{2}-21 \times \frac{22}{7} \times(0.7)^{2} \\
& =182.34 \mathrm{~cm}^{2}
\end{aligned}
\]
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