Question

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

Answer 1

Hint:
To find out the relation between the height and radius of the cylinder, we just need to follow the question and take the ratio of curved surface area and total surface area of the cylinder as we will get the relation.

Complete step by step solution:
Given
Ratio between the curved surface area and the total surface area of a right circular cylinder \[ = 1:2\]
Let the radius of the cylinder $ = r$
And, height of the cylinder $ = h$
Curved Surface area of a cylinder: The cylinder is a three-dimensional shape, its curved surface area is the area of the only curved surface.
Total surface area of cylinder: As the cylinder is three-dimensional solid shape its total surface area of the cylinder is equal to the sum of curved surface area and two circular bases of the cylinder.
As we know that formula for curved surface area of cylinder$ = 2\pi rh$
And, total surface area of the cylinder $ = 2\pi rh + 2\pi {r^2}$
As per the question ratio between the curved surface area and the total surface area of a right circular cylinder is\[1:2\], so we can write this in equation as below:
$\dfrac{{2\pi rh}}{{2\pi rh + 2\pi {r^2}}} = \dfrac{1}{2}$
We will take $2\pi r$common
$\dfrac{{2\pi rh}}{{2\pi r(h + r)}} = \dfrac{1}{2}$
$2\pi r$will be cancel out
$\dfrac{h}{{r + h}} = \dfrac{1}{2}$
After applying cross-multiplication
$2h = r + h$
Take h on left side and remaining on right side
$h = r$
$\dfrac{h}{r} = \dfrac{1}{1}$

Hence the ratio between radius and height of the cylinder is $1:1$.

Note:
We keep in mind that the total surface area of the cylinder is equal to the sum of the curved surface area of the cylinder and the area of two circles, one is the top of the cylinder and another one at the bottom of the cylinder.