Question

The radii of two cylinders are in the ratio \[3:2\] and their heights are in the ratio \[4:5\]. Calculate the ratio of their curved surface areas.

Answer 1

Hint: In this question, we first need to know the formula for the curved surface area of a cylinder. Then write the ratio of the curved surface area of the two cylinders and then substitute the ratio of radii and heights in it. Now, on simplifying it further we get the result.

Complete step-by-step answer:
Now, let us assume the curved surface areas of the two cylinders as $A_1$ and $A_2$ and their respective radius as $r_1$ and $r_2$ and the corresponding heights as $h_1$ and $h_2$.
Solid Figure:
The objects which occupy space ( i.e. they have three dimensions) are called solids.
Right Circular Cylinder:
A right circular cylinder is considered as a solid generated by the revolution of a rectangle about one of its sides.
Let r be the radius and h be the height of the cylinder
Then the curved surface area of the cylinder is given by the formula
\[\Rightarrow 2\pi rh\]

Now, from the given conditions in the question we have
\[\begin{align}
  & {{r}_{1}}:{{r}_{2}}=3:2 \\
 & {{h}_{1}}:{{h}_{2}}=4:5 \\
\end{align}\]
Now, from the formula of curved surface area of a cylinder we get,
\[\Rightarrow A=2\pi rh\]
Now, on substituting the respective values we get,
\[\begin{align}
  & \Rightarrow {{A}_{1}}=2\pi {{r}_{1}}{{h}_{1}} \\
 & \Rightarrow {{A}_{2}}=2\pi {{r}_{2}}{{h}_{2}} \\
\end{align}\]
Now, the ratio of the curved surface area can be written as
\[\Rightarrow {{A}_{1}}:{{A}_{2}}=2\pi {{r}_{1}}{{h}_{1}}:2\pi {{r}_{2}}{{h}_{2}}\]
Now, on cancelling out the common terms we get,
\[\Rightarrow {{A}_{1}}:{{A}_{2}}={{r}_{1}}{{h}_{1}}:{{r}_{2}}{{h}_{2}}\]
Now, on substituting the respective ratio values we get,
\[\Rightarrow {{A}_{1}}:{{A}_{2}}=\dfrac{3}{2}\times \dfrac{4}{5}\]
Now, on cancelling out the common terms we get,
\[\Rightarrow {{A}_{1}}:{{A}_{2}}=\dfrac{3}{1}\times \dfrac{2}{5}\]
Now, on further simplification we get,
\[\therefore {{A}_{1}}:{{A}_{2}}=6:5\]

Note: It is important to note that we need to find the curved surface area not the total surface area because instead of considering the formula of curved surface area if we consider the total surface area then the result will be completely different.
It is also to be noted that while substituting the values we need to consider the corresponding ratios because interchanging the numerator and denominator changes the result.