Question

The total surface area of a right circular cylinder is $165\pi c{{m}^{2}}$. If the radius of its base is 5 cm; find its height and volume.

Answer 1

Hint: Total surface area of the cylinder is given as $2\pi rh+2\pi {{r}^{2}}$, where ‘r’ is the radius of the base and ‘h’ is the height of the cylinder and value of $\pi $ is 3.14. Volume of the cylinder is given as $\pi {{r}^{2}}h$. Find the volume by calculating the height from the given total surface area in the problem and use the above mentioned formulae.

Complete step by step answer:
We have a right circular cylinder with a total surface area of $165\pi c{{m}^{2}}$ and radius of base as 5 cm and hence, we need to determine the value of height and volume of the cylinder.

Let the height of the cylinder be h cm.
As we know the total surface area of the cylinder can be given as $2\pi rh+2\pi {{r}^{2}}$ which includes $2\pi rh$ for the curved surface area and $2\pi {{r}^{2}}$ for the areas of both the circles of the bases (upper and lower circle).
Hence, we have
Total surface area $=2\pi rh+2\pi {{r}^{2}}$
And we have a total surface area as $165\pi c{{m}^{2}}$ from the problem. So, we get
$2\pi rh+2\pi {{r}^{2}}=165\pi $
$\Rightarrow 2\pi r\left( h+r \right)=165\pi $
$2r\left( h+r \right)=165$
We have $r=5cm$ from the question (radius of cylinder). Hence, we can get the above equation as
$2\times 5\left( h+5 \right)=165$
$10\left( h+5 \right)=165$
$10h+50=165$
$10h=165-50=115$
$h=\dfrac{115}{10}=11.5cm$
Now, we can calculate volume of the cylinder with the help of the relation given as
Volume of cylinder $=\pi {{r}^{2}}h$
Where r= radius
h=height
and $\pi =\dfrac{22}{7}=3.14$
 So, we get
Volume of the cylinder given $=\pi \times {{\left( 5 \right)}^{2}}\times 11.5$
$=\pi \times 25\times 11.5$
$=3.14\times 25\times 11.5$
$=902.75c{{m}^{3}}$
Hence, height and volume of the right circular cylinder are 11.5 cm and $902.75c{{m}^{3}}$ respectively.

Note: Don’t put the total surface of the cylinder as $2\pi rh$ . It is the curved surface area of the cylinder, we need to include the area of circles with the bases (upper and lower circles). So, total surface area can be given as
$TSA=CSA+$ Area of bases
$TSA=2\pi rh+2\pi {{r}^{2}}$
And don’t confuse between total surface area and volume as well.
Volume of the cylinder is given as $\pi {{r}^{2}}h$.
Put the values of ‘r’ and ‘h’ carefully in the identities of volume and surface area. Don’t change the place of them while putting the value of radius and height. Be careful with it as well.