Answer: The correct option is (A) 125 : 64.
Step-by-step explanation: Given that the ratio of the heights and radii of two similar cylinders is 5 : 4.
We are to find the ratio for the volumes of the two cylinders.
We know that the volume of a cylinder with radius r units and height h units is given by
![V=\pi r^2h.](https://tex.z-dn.net/?f=V%3D%5Cpi%20r%5E2h.)
Let r, r' be the radii and h, h' be the heights of the two similar cylinders.
Then, the volumes of the two cylinders will be
![V=\pi r^2h,\\\\\\V'=\pi r'^2h'.](https://tex.z-dn.net/?f=V%3D%5Cpi%20r%5E2h%2C%5C%5C%5C%5C%5C%5CV%27%3D%5Cpi%20r%27%5E2h%27.)
According to the given information, we have
![\dfrac{r}{r'}=\dfrac{h}{h'}=\dfrac{5}{4}.](https://tex.z-dn.net/?f=%5Cdfrac%7Br%7D%7Br%27%7D%3D%5Cdfrac%7Bh%7D%7Bh%27%7D%3D%5Cdfrac%7B5%7D%7B4%7D.)
Therefore, we get
![\dfrac{V}{V'}=\dfrac{\pi r^2h}{\pi r'^2h'}=\left(\dfrac{r}{r'}\right)^2\times\dfrac{h}{h'}=\left(\dfrac{5}{4}\right)^2\times\dfrac{5}{4}=\dfrac{125}{64}=125:64.](https://tex.z-dn.net/?f=%5Cdfrac%7BV%7D%7BV%27%7D%3D%5Cdfrac%7B%5Cpi%20r%5E2h%7D%7B%5Cpi%20r%27%5E2h%27%7D%3D%5Cleft%28%5Cdfrac%7Br%7D%7Br%27%7D%5Cright%29%5E2%5Ctimes%5Cdfrac%7Bh%7D%7Bh%27%7D%3D%5Cleft%28%5Cdfrac%7B5%7D%7B4%7D%5Cright%29%5E2%5Ctimes%5Cdfrac%7B5%7D%7B4%7D%3D%5Cdfrac%7B125%7D%7B64%7D%3D125%3A64.)
Thus, the required ratio of the volumes of the two cylinders is 125 : 64.
Option (A) is CORRECT.