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 depressions and so on. The sum of number of hemispheric

Question

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 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 5 cm and each hemispherical depression is of diameter 1.4 cm, find the total surface area of the die so formed.

Accepted Answer

Step 1: Number of hemispherical depressions $= 1 + 2 + 3 + 4 + 5 + 6 = 21$
Step 2: Total surface area of the die formed
$=$ TSA of cube $+$ CSA of 21 hemispheres $-$ Base area of 21 hemispheres
Step 3: $= 6 	imes 5^2 + 21 	imes 2 	imes \frac{22}{7} 	imes (0.7)^2 - 21 	imes \frac{22}{7} 	imes (0.7)^2$
Step 4: $= 150 + 64.68 - 32.34$
Step 5: $= 182.34$ cm$^2$

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