Question

The ratio between the curved surface area and the total surface area of a cylinder is 1:2. Find the ratio between the height and the radius of the cylinder.

Answer 1

Hint: We will use the formula of the total surface of the cylinder as \[2\pi r(h+r)\]and the formula of Curved surface area of the cylinder as\[2\pi rh\], where r is the radius of the cylinder and h is the height of the cylinder.

Complete step-by-step answer:

Given that the ratio between the curved surface area and the total surface area of a cylinder is 1:2 and we have to find the ratio of the height and radius of the cylinder.
Let height of the cylinder be h and the radius of the cylinder be ‘r’,
The formula of total surface of the cylinder is \[2\pi r(h+r)\]and the formula of Curved surface area of the cylinder is\[2\pi rh\], where r is the radius of the cylinder and h is the height of the cylinder.

Therefore, according to the given in the question,
\[\dfrac{\text{Curved surface area}}{\text{Total surface area}}=\dfrac{2\pi rh}{2\pi r(h+r)}=\dfrac{1}{2}\]
\[\Rightarrow \dfrac{2\pi rh}{2\pi r(h+r)}=\dfrac{1}{2}\]
Cancelling the common terms, we get
\[\Rightarrow \dfrac{h}{(h+r)}=\dfrac{1}{2}\]
By cross multiplication,
\[\Rightarrow 2h=h+r\]
\[\Rightarrow h=r\]
Therefore, by calculating we get height is equal to radius.
Therefore, the ratio of height and radius of the cylinder will be 1:1.

Note: The possibility of error in the question is obtaining the value of radius of the cylinder and the height of the cylinder separately and not directly going for the ratio of both the radius and the height of the cylinder.