Question
The volume of two spheres are in the ratio of 64 : 27. The ratio p their surface areas is :
Answer: Option D
Let their radii be R and r
Then,
$$\eqalign{
& \frac{{\frac{4}{3}\pi {R^3}}}{{\frac{4}{3}\pi {r^3}}} = \frac{{64}}{{27}} \cr
& \Rightarrow {\left( {\frac{R}{r}} \right)^3} = \frac{{64}}{{27}} \cr
& \Rightarrow {\left( {\frac{R}{r}} \right)^3} = {\left( {\frac{4}{3}} \right)^3} \cr
& \Rightarrow \frac{R}{r} = \frac{4}{3} \cr} $$
Ratio of surface areas :
$$\eqalign{
& = \frac{{4\pi {R^2}}}{{4\pi {r^2}}} = {\left( {\frac{R}{r}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {\frac{4}{3}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{16}}{9} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 16:9 \cr} $$
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Let their radii be R and r
Then,
$$\eqalign{
& \frac{{\frac{4}{3}\pi {R^3}}}{{\frac{4}{3}\pi {r^3}}} = \frac{{64}}{{27}} \cr
& \Rightarrow {\left( {\frac{R}{r}} \right)^3} = \frac{{64}}{{27}} \cr
& \Rightarrow {\left( {\frac{R}{r}} \right)^3} = {\left( {\frac{4}{3}} \right)^3} \cr
& \Rightarrow \frac{R}{r} = \frac{4}{3} \cr} $$
Ratio of surface areas :
$$\eqalign{
& = \frac{{4\pi {R^2}}}{{4\pi {r^2}}} = {\left( {\frac{R}{r}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {\frac{4}{3}} \right)^2} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{16}}{9} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 16:9 \cr} $$
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