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
[tex]V=\pi r^2h.[/tex]
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
[tex]V=\pi r^2h,\\\\\\V'=\pi r'^2h'.[/tex]
According to the given information, we have
[tex]\dfrac{r}{r'}=\dfrac{h}{h'}=\dfrac{5}{4}.[/tex]
Therefore, we get
[tex]\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.[/tex]
Thus, the required ratio of the volumes of the two cylinders is 125 : 64.
Option (A) is CORRECT.
