Question

Consider a hollow cylinder of inner radius $r$ and thickness of wall $t$ and length $l$ . The volume of the above cylinder is given by
A). $2\pi lt({r^2} - {l^2})$
B). $2\pi rlt(\dfrac{t}{{2r}} + 1)$
C). $2\pi lt({r^2} + {l^2})$
D). $2\pi rlt(r + t)$

Answer 1

Hint: Let’s consider a hollow cylinder with the inner side radius is given as $r$. So, the radius can find with the help of diameter, which is half of the diameter is the radius and represented as $r = \dfrac{d}{2}$
And the Thickness of the wall is given as $t$and also, we have their length as $l$
Then the requirement that we find the volume of the cylinder.

Formula used:
Volume of cylinder $V = \pi {r^2}h$

Complete step-by-step solution:
Now, we need to find the volume of the above cylinder
We know that,
Volume of cylinder $V = \pi {r^2}h$
Let us consider
$V = \pi {r^2}l$ Because we have that height is equal to the length in the cylinder
Now we apply for the general formula in the condition
We know that,
The total volume of the cylinder with radius (r) + thickness (t) = \[r{\text{ }} + {\text{ }}t\]
Therefore, Required volume = Total volume of the cylinder with radius (r) + thickness (t) – the volume of the cylinder with radius (r)
Let us solve the equation, =$\pi {(r + t)^2}l - \pi {r^2}l$ (volume of the cylinder)
So, we on simplifying,
We have to take the common terms in the above equation
=$l[\pi {(r + t)^2} - \pi {r^2}]$
We see the bracket${(r + t)^2}$
So, we apply the formula in ${(r + t)^2}$
\[{\left( {r{\text{ }} + {\text{ }}t} \right)^2}\; = {\text{ }}{r^2} + {\text{ }}{t^2} + 2rt\]
So, apply the formula in the above equation
=$\pi l[{r^2} + {t^2} + 2rt - {r^2}]$
We cancel the\[{r^2}\]
Because opposite sign will occur like \[\left( {{\text{ }}{r^2} - {\text{ }}{r^2}} \right){\text{ }} = {\text{ }}0\]
Now we get
$ = \pi l[{t^2} + 2rt]$
Then we take the common term in the bracket
t is the common term
$ = \pi lt[t + 2r]$
We take the \[2r\] in the bracket
Then we on simplifying,
$ = 2r\pi lt[\dfrac{t}{{2r}} + 1]$
Hence, the correct option is (B) $2\pi rlt(\dfrac{t}{{2r}} + 1)$

Note: Volume is space occupied by a three-dimensional object. then the volume of the cuboid = base area $\times$ height.
Now, let us deal with the cylinder, just like the cuboid, we can see that top and base areas are parallel to each other. We see that both the top and bottom circles have the same radii. so both circles’ areas are congruent to each other. And, the lateral area is perpendicular to the base. Therefore, we can multiply the base area by height, to find out the volume.