Question
The ratio of the volumes of a right circular cylinder and a sphere is 3 : 2. If the radius of the sphere is double the radius of the base of the cylinder, find the ratio of the total surface areas of the cylinder and the sphere :
Answer: Option D
Let the radius of the cylinder be r
Then, radius of the sphere = 2r
$$\eqalign{
& \frac{{{\text{Volume of cylinder}}}}{{{\text{Volume of sphere}}}} = \frac{3}{2} \cr
& \Rightarrow \frac{{\pi {r^2}h}}{{\frac{4}{3}\pi {{\left( {2r} \right)}^3}}} = \frac{3}{2} \cr
& \Rightarrow \frac{h}{r} = 16 \cr
& \Rightarrow h = 16r \cr} $$
∴ Required ratio :
$$\eqalign{
& \frac{{{\text{Total surface area of cylinder}}}}{{{\text{Surface area of sphere}}}} \cr
& = \frac{{2\pi r.\left( {16r} \right) + 2\pi {r^2}}}{{4\pi {{\left( {2r} \right)}^2}}} \cr
& = \frac{{34\pi {r^2}}}{{16\pi {r^2}}} \cr
& = \frac{{17}}{8}\,Or\,17:8 \cr} $$
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Let the radius of the cylinder be r
Then, radius of the sphere = 2r
$$\eqalign{
& \frac{{{\text{Volume of cylinder}}}}{{{\text{Volume of sphere}}}} = \frac{3}{2} \cr
& \Rightarrow \frac{{\pi {r^2}h}}{{\frac{4}{3}\pi {{\left( {2r} \right)}^3}}} = \frac{3}{2} \cr
& \Rightarrow \frac{h}{r} = 16 \cr
& \Rightarrow h = 16r \cr} $$
∴ Required ratio :
$$\eqalign{
& \frac{{{\text{Total surface area of cylinder}}}}{{{\text{Surface area of sphere}}}} \cr
& = \frac{{2\pi r.\left( {16r} \right) + 2\pi {r^2}}}{{4\pi {{\left( {2r} \right)}^2}}} \cr
& = \frac{{34\pi {r^2}}}{{16\pi {r^2}}} \cr
& = \frac{{17}}{8}\,Or\,17:8 \cr} $$
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