The surface area of a sphere of radius 4 cm is eight times the area of the curved surface area of a cone whose radius 2 cm. Find the volume of the cone.

Correct option is C

Given:

Radius of sphere = 4 cm

Radius of base of cone = 2 cm

Surface area of sphere is 8 times the curved surface area of cone

Formula Used:

Curved surface area of cone = $$\pi r l$$​​

Slant height , $$l= \sqrt{r^{2}+h^{2}}$$​​

Surface area of sphere = $$4 \pi r^{2}$$​​

Volume of cone = $$\frac{1}{3}\times\pi \times r^{2}\times h$$​​

Solution:

Surface area of sphere = $$4 \pi 4^{2} = 64\pi \ cm^{2}$$​​

Surface area of sphere is 8 times the curved surface area of cone

Curved surface area of cone = $$\frac{1}{8} \times 64\pi \ cm^{2} = 8\pi \ cm^{2}$$​​

​$$\pi r l = 8\pi \ cm^{2}$$​ ( radius of base of cone = 2cm)

​$$l= 4 \ cm^{2}$$​​

​$$l= \sqrt{r^{2}+h^{2}} = 4$$​​

Squaring both sides

​$$2^{2}+h^{2} = 16$$​​

​$$h^{2}= 16-4 =12$$​​

​$$h=2\sqrt{3} \ cm$$​​

Volume of cone=​​​$$=\frac{1}{3}\times\pi \times 2^{2}\times 2\sqrt{3}\ cm$$​​

​$$= \frac{8\pi}{\sqrt{3}} \ cm^{3}$$​​

Thus, Volume of cone = $$\bf \frac{8\pi}{\sqrt{3}}\ cm^{3}$$​​