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.

Answer 1

Number of depressions on each face are 1, 2, 3, 4, 5, 6 respectively. 

CSA of each depression = 2\(\pi\)r²

^{= \(2 × \frac{22}{7} ×0.7×0.7\)}

^{= \(44×0.1×0.7\)}   

^{= 3.08 \(cm^2\)}   

Base area of hemisphere = \(\pi r^2\) = \(\frac{22}{7} × 0.7 ×0.7 = 1.54 cm^2\)   

TSA of cube = 6 × 5² = 150 cm²

TSA of the die = 150 + (1 + 2 + 3 + 4 + 5 + 6) × 3.08 – (1 + 2 + 3 + 4 + 5 + 6) × 1.54

= 150 + 21 × 3.08 – 21 × 1.54

= 150 + 64.68 – 32.34

= 182.34 cm²