Answer: The ratio of the volumes of the two cylinders is 512 : 3375.
Step-by-step explanation: Given that two similar cylinders have a scale factor of [tex]\dfrac{1.6}{3}.[/tex]
We are to find the ratio of 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 and R be the radii and h and H be the heights of the two cylinders.
Then, we must have
[tex]\dfrac{R}{r}=\dfrac{1.6}{3}=\dfrac{16}{30}=\dfrac{8}{15}\\\\\Rightarrow R=\dfrac{8}{15}r.[/tex]
And, similarly,
[tex]H=\dfrac{8}{15}h.[/tex]
Therefore, the volume of the first cylinder will be
[tex]V_1=\pi r^2h[/tex]
and volume of the second cylinder will be
[tex]V_2=\pi R^2H=\pi\times\left(\dfrac{8}{15}r\right)^2\times\left(\dfrac{8}{15}h\right)=\dfrac{512}{3375}\pi r^2h.[/tex]
Thus, the required ratio of the volumes of the two cylinders is given by
[tex]\dfrac{V_2}{V_1}=\dfrac{\frac{512}{3375}\pi r^2h}{\pi r^2h}=\dfrac{512}{3375}=512:3375.[/tex]
The ratio of the volumes of the two cylinders is 512 : 3375.
