The ratio of total surface area and volume of a sphere is 1 : 7. This sphere is melted to form small spheres of equal size. The radius of each small sphere is $$\frac{1}{6}$$ the radius of the large sphere. What is the sum (in cm²) of curved surface areas of small spheres?

Solution (By Examveda Team)

$$\eqalign{ & \frac{{{\text{Total surface area}}}}{{{\text{Volume}}}} = \frac{1}{7} \cr & \frac{{4\pi {r^2}}}{{\frac{4}{3}\pi {r^3}}} = \frac{1}{7} \cr & r = 21{\text{ cm}} \cr & {\text{Radius of small sphere}} = \frac{1}{6} \times r \cr & = \frac{1}{6} \times 21 \cr & = \frac{7}{2}{\text{ cm}} \cr & {\text{Number of small sphere}} \cr & = \frac{{{\text{Volume of large sphere}}}}{{{\text{Volume of small sphere}}}} \cr & = \frac{{\frac{4}{3}\pi {R^3}}}{{\frac{4}{3}\pi {r^3}}} \cr & = \frac{{21 \times 21 \times 21}}{{\frac{7}{2} \times \frac{7}{2} \times \frac{7}{2}}} \cr & = 27 \times 8 \cr & = 216 \cr & {\text{Curved surface area of small sphere}} \cr & = 216 \times 4\pi {r^2} \cr & = 216 \times 4 \times \frac{{22}}{7} \times \frac{7}{2} \times \frac{7}{2} \cr & = 33264{\text{ c}}{{\text{m}}^2} \cr} $$