Question

The radius and height of a cylinder are in the ratio $ 5:7 $ and its curved surface area is $ 5500\;{\text{sq}}{\text{. cm}} $ find its radius and height.

Answer 1

Hint: The curved surface area of a cylinder is defined as the area of a surface of the cylinder along the curve of the body of the cylinder. It does not include the two circular ends of the cylinder.

Complete step-by-step answer:
Let us assume the radius and the height of a cylinder be $ r{\text{ and h}} $ respectively.
The ratio of the radius and height of the cylinder given is $ 5:7 $ so,
 $ \begin{array}{c}
\dfrac{r}{h} = \dfrac{5}{7}\\
{\text{or }}\dfrac{5}{7} = x\left( {{\text{suppose}}} \right){\text{ }}
\end{array} $
Then, the value of the radius and height of the cylinder will be $ 5x{\text{ and }}7x $ .
Also given, the curved surface area of the cylinder,
 $ {A_{cs}} = 5500{\text{ c}}{{\text{m}}^2} $
We know the formula for the curved surface area of the cylinder is given by,
 $ {A_{cs}} = 2\pi rh $
Substituting this into the formula we get,
 $ 2\pi rh = 5500{\text{ c}}{{\text{m}}^2} $
Now substituting the value of radius $ r = 5x $ and height $ h = 7x $ we get,
 $ \begin{array}{l}
2 \times \dfrac{{22}}{7} \times 5x \times 7x = 5500{\text{ c}}{{\text{m}}^2}\\
\dfrac{{44}}{7} \times 35{x^2} = 5500{\text{ c}}{{\text{m}}^2}\\
{x^2} = \dfrac{{5500 \times 7}}{{44 \times 35}}
\end{array} $
Solving this we get,
 $ {x^2} = 25 $
Taking square root of both sides we get,
 $ x = 5 $
So, the value of $ x $ calculated is $ 5 $ . Now using this value, we can calculate the radius and the height of the cylinder.
We know that the radius of the cylinder id given by-
 $ r = 5x $
Substituting $ x = 5 $ we get,
 $ \begin{array}{c}
r = 5 \times 5\\
r = 25
\end{array} $
So, the radius of the cylinder is \[25\;{\text{cm}}\].
Similarly, the height of the cylinder is given by-
 $ h = 7x $
Substituting $ x = 5 $ we get,
 $ \begin{array}{c}
h = 7 \times 5\\
h = 35
\end{array} $
So, the height of the cylinder is $ 35\;{\text{cm}} $ .

Therefore, the radius and the height of the cylinder are \[25\;{\text{cm}}\] and $ 35\;{\text{cm}} $ respectively.

Note: An alternative method of solving this question is given by-
Given the ratio between the cylinder radius $ r $ and height $ h $ is,
 $ \begin{array}{c}
\dfrac{r}{h} = \dfrac{5}{7}\\
{\text{or, }}h = \dfrac{7}{5}r
\end{array} $
Now using the formula for the curved surface area,
 $ \begin{array}{c}
{A_{cs}} = 2\pi rh\\
5500{\text{c}}{{\text{m}}^2} = 2\pi rh\left( {{A_{cs}} = 5500{\text{c}}{{\text{m}}^2}{\text{ is given}}} \right)
\end{array} $
Substituting $ h = \dfrac{7}{5}r $ , we get,
 $ \begin{array}{c}
5500 = 2 \times \dfrac{{22}}{7}r \times \dfrac{7}{5}r\\
\dfrac{{5500 \times 35}}{{44 \times 7}} = {r^2}
\end{array} $
Taking square root of both sides and solving we get,
 $ r = 25 $
And substituting this value of $ r $ in the relation $ h = \dfrac{7}{5}r $ we get,
 $ \begin{array}{c}
h = \dfrac{7}{5} \times 25\\
h = 35
\end{array} $
So, the radius and the height of this cylinder is $ 25\;{\text{cm and 35}}\;{\text{cm}} $ respectively.