Answer: The required similarity ratio of the smaller to the larger similar cylinder is
a : b = 4 : 5.
Step-by-step explanation: We are given to find the similarity ratio of the smaller to the larger similar cylinder.
Let S and S' be the surface area of the smaller and bigger cylinders respectively.
Then, according to the given information, we have
[tex]S=48\pi~\textup{m}^2,\\\\S'=75\pi~\textup{m}^2.[/tex]
We know that
Similarity ratio of two cylinders is the ratio of the radii of the two cylinders and the ratio of the surface area of the two cylinders is the square of the similarity ratio.
Therefore, if a and b represents the radius of the smaller and bigger cylinder respectively, then
[tex]\dfrac{S}{S'}=\dfrac{48\pi}{75\pi}\\\\\\\Rightarrow \dfrac{a^2}{b^2}=\dfrac{16}{25}\\\\\\\Rightarrow \dfrac{a^2}{b^2}=\dfrac{4^2}{5^2}\\\\\\\Rightarrow \left(\dfrac{a}{b}\right)^2=\left(\dfrac{4}{5}\right)^2\\\\\\\Rightarrow \dfrac{a}{b}=\dfrac{4}{5}\\\\\Rightarrow a:b=4:5.[/tex]
Thus, the required similarity ratio of the smaller to the larger similar cylinder is
a : b = 4 : 5.
