Question

A solid cylinder has a total surface area of $462c{{m}^{2}}$ . Its curved surface area is one third of its total surface area. Find the volume of the cylinder. (Take $\pi =\dfrac{22}{7}$ ).
A. 224
B. 845
C. 664
D. None of the above

Answer 1

Hint: Use the formula of total surface area which is $2\pi r\left( h+r \right)$ and curved surface area is $2\pi rl$. Then use the relation to get the relation between radius and height and hence, find the volume.

Complete step-by-step answer:
In the question, we are given that the total surface area of the solid cylinder is $462c{{m}^{2}}$. We are also said that its curved surface area is one-third of its total surface area, hence, we have to find the volume of the cylinder.

Now, as we know the relation between curved surface area and total surface area, we can use it to find relation between height and radius of cylinder.
As we know the curved surface area of the cylinder is $2\pi rh$ and total surface area is $2\pi r\left( h+r \right)$ where h is the height of the cylinder and r is the radius. So, we can write the equation according to question as,
$2\pi rh=\dfrac{1}{3}2\pi r\left( h+r \right)$
Now, we will multiply 3 to both sides of the equation. We get
$6\pi rh=2\pi r\left( h+r \right)$
Now, we will divide the equation by $2\pi r$ both sides. We get,
$3h=h+r$
Now, subtracting ‘h’ from both sides. We get,
$r=2h$
In the question, we are given the value of the total surface area of the cone as $462c{{m}^{2}}$ . So, we can write $2\pi r\left( h+r \right)$ equals to 462 where r is radius, h is height and $\pi =\dfrac{22}{7}$ .
$2\pi rh\left( h+r \right)=462$
We found the relation that $r=2h$ . So,
$2\pi \left( 2h \right)\left( h+2h \right)=462$
On further simplification, we can write it as,
$4\pi h\times 3h=462$
Now putting $\pi =\dfrac{22}{7}$ we get,
$12\times \dfrac{22}{7}\times {{h}^{2}}=462$
On doing calculation we get,
${{h}^{2}}=12.25$
So, $h=3.5cm$
Now, as we know $r=2h$ so, $r=2\times 3.5cm$ $\Rightarrow r=7cm$
We know that the radius of the cylinder is 7 cm and height is 3.5cm.
Now we have to find the volume which we get by using the formula $\pi {{r}^{2}}h$ ,where r is radius and h is the height.
$Volume=\pi {{r}^{2}}h=\dfrac{22}{7}\times {{\left( 7 \right)}^{2}}\times 3.5$
Hence, on calculation we get
$Volume=539c{{m}^{3}}$
So, the correct option is ‘D’.

So, the correct answer is “Option D”.

Note: Students generally tend to forget the formula of total and curved surface area of various 3-D figures. They should be careful while doing calculations to avoid any mistakes. Total surface area is different from curved surface area, in curved surface area the two ends of the cylinder are not considered. Students often get confused here.