Question

Find the volume of the cylinder whose total surface area is $1628{\text{ c}}{{\text{m}}^2}$ and the circumference of its cross-section is $44{\text{ cm}}$.

Answer 1

Hint: The cross-section of the cylinder is circular in shape. Use the formula $2\pi r$ for the circumference of the cross-section of the cylinder to find its radius. Then use the formula of total surface area of the cylinder i.e. $2\pi r\left( {h + r} \right)$ to determine its height. Finally put these values in the formula of volume of cylinder i.e. $\pi {r^2}h$ to get the answer.

Complete step-by-step answer:
According to the question, the total surface area of the cylinder is $1628{\text{ c}}{{\text{m}}^2}$.
We know that the total surface area of a cylinder is $2\pi r\left( {h + r} \right)$, where $r$ is the radius of its cross-section and $h$ is its height. Using this formula, we’ll get:
\[ \Rightarrow 2\pi r\left( {h + r} \right) = 1628{\text{ }}....{\text{(1)}}\]
Further, the circumference of its cross-section is given as $44{\text{ cm}}$.
We also know that the cross-section of a cylinder is circular in shape. So the circumference of the cross-section will be $2\pi r$. Using this, we’ll get:
\[ \Rightarrow 2\pi r = 44\]
Now, putting $\pi = \dfrac{{22}}{7}$, we’ll get:
\[
   \Rightarrow 2 \times \dfrac{{22}}{7} \times r = 44 \\
   \Rightarrow r = 7{\text{ cm}}
 \]
Putting the value of $r$ in equation (1) and using $\pi = \dfrac{{22}}{7}$, we’ll get:
\[
   \Rightarrow 2 \times \dfrac{{22}}{7} \times 7 \times \left( {h + 7} \right) = 1628 \\
   \Rightarrow \left( {h + 7} \right) = \dfrac{{1628}}{{44}} = 37 \\
   \Rightarrow h = 30{\text{ cm}}
 \]
Now, we know that the volume of a cylinder is given by the formula:
$ \Rightarrow V = \pi {r^2}h$
Putting the values of $r$ and $h$ calculated above and again using $\pi = \dfrac{{22}}{7}$, we’ll get:
$
   \Rightarrow V = \dfrac{{22}}{7} \times 7 \times 7 \times 30 \\
   \Rightarrow V = 22 \times 7 \times 30 \\
   \Rightarrow V = 4620{\text{ c}}{{\text{m}}^3}
 $

Thus the volume of the cylinder is $4620{\text{ c}}{{\text{m}}^3}$.

Note: The total surface area of a cylinder comprises its curved surface area and the area of its two cross-sections. The curved surface area is given by the formula:
$ \Rightarrow $ Curved Surface Area of Cylinder $ = 2\pi rh$.
And the area of its cross-section is given by the formula:
$ \Rightarrow $ Cross-Sectional Area of Cylinder $ = \pi {r^2}$.
But since there are two cross-sections in a cylinder, so we have:
$ \Rightarrow $ Total Cross-Sectional Area of Cylinder $ = 2\pi {r^2}$.
On adding these two areas, we get the total surface area of a cylinder, which we have used above.
$ \Rightarrow $ Total Surface Area of Cylinder $ = 2\pi rh + 2\pi {r^2} = 2\pi r\left( {h + r} \right)$.