Question

Which of the following expressions represents the ratio of the surface area of a cylinder to the surface area of a sphere with the same radius, r?

Answer 1

The ratio of the surface area of a cylinder to the surface area of a sphere is $\frac{r+h}{2r}$, that is mentioned in option E.

Step-by-step explanation:

The given is,

                Cylinder and sphere has same area.

Step: 1

               Formula for surface area of cylinder,

                                 $A = 2\pi rh + 2\pi r^{2}$.............................(1)

               Where, r - Radius of cylinder

                           h - Height of cylinder

Step:2

              Formula for surface area of sphere,

                                $A=4\pi r^{2}$.......................................(2)

              Where, r - Radius of sphere

Step:3

             Ratio of the surface area of a cylinder to the surface area of a sphere,

                                     $= \frac{Surface area of cylinder}{Surface area of sphere}$

                                     $=\frac{ 2\pi rh + 2\pi r^{2} }{ 4\pi r^{2}}$

                                     $= \frac{2\pi r h }{ 4\pi r^{2}} + \frac{ 2\pi r^{2} }{4\pi r^{2}}$

                                     $= \frac{ h }{ 2 r} + \frac{ 1 }{2}$

                                     $= \frac{2h + 2r}{(2r)(2)}$

                                     $= \frac{2(r + h)}{2 (2r)}$

                                     $= \frac{r + h}{2r}$

              Ratio of the surface area of a cylinder to the surface area of a sphere   $= \frac{r + h}{2r}$ .

Result:

              The ratio of the surface area of a cylinder to the surface area of a sphere is $\frac{r+h}{2r}$, that is mentioned in option E.