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 \(\dfrac{1.6}{3}.\)
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
\(V=\pi r^2h.\)
Let r and R be the radii and h and H be the heights of the two cylinders.
Then, we must have
\(\dfrac{R}{r}=\dfrac{1.6}{3}=\dfrac{16}{30}=\dfrac{8}{15}\\\\\Rightarrow R=\dfrac{8}{15}r.\)
And, similarly,
\(H=\dfrac{8}{15}h.\)
Therefore, the volume of the first cylinder will be
\(V_1=\pi r^2h\)
and volume of the second cylinder will be
\(V_2=\pi R^2H=\pi\times\left(\dfrac{8}{15}r\right)^2\times\left(\dfrac{8}{15}h\right)=\dfrac{512}{3375}\pi r^2h.\)
Thus, the required ratio of the volumes of the two cylinders is given by
\(\dfrac{V_2}{V_1}=\dfrac{\frac{512}{3375}\pi r^2h}{\pi r^2h}=\dfrac{512}{3375}=512:3375.\)
The ratio of the volumes of the two cylinders is 512 : 3375.
