A solid brass sphere of radius 21 cm is converted into a right circular cylindrical rod of length 28 cm. The ratio of total surface areas of the rod to sphere is :

Since, the solid brass sphere is converted to right circular cylindrical rod, there volumes must be same

Now volume of sphere = $$\dfrac{4}{3}\pi\ r^3$$ = $$\dfrac{4}{3}\pi\ \left(21\right)^3$$

Volume of cylinder = $$\pi\ r^2l$$ = $$\pi\ r^2\left(28\right)$$

So, $$\dfrac{4}{3}\pi\ \left(21\right)^3$$ = $$\pi\ r^2\left(28\right)$$

or, $$r^2=441$$

or, $$r=21$$ cm

So, basically we can see radius of both the cylinder and sphere are same.

So, total surface area of rod = $$2\pi\ rl+2\pi\ r^2$$

Also, total surface area of sphere = $$4\pi\ r^2$$

So, required ratio = $$\dfrac{2\pi rl+2\pi\ r^2}{4\pi\ r^2}=\dfrac{l+r}{2r}=\dfrac{21+28}{2\times\ 21}=\dfrac{49}{42}=\dfrac{7}{6}$$