Question

The TSA of a solid cylinder is $231c{m^2}$ and the CSA is $\dfrac{2}{3}$ of TSA. Find the volume of the cylinder.

Answer 1

Hint: According to the question given in the question we have to determine the volume of the cylinder when TSA of solid cylinder is $231c{m^2}$ and the CSA is $\dfrac{2}{3}$ of TSA. So, first of all we have to use the formula to determine the total surface area (TSA) of the solid cylinder which is as given below:

Formula used: $ \Rightarrow $TSA$ = 2\pi r(h + r)$…………………(A)
Where, r is the radius and h is the height of the solid cylinder.
So, with the help of the formula (A) to find the TSA of the solid cylinder we can determine the CSA of the given solid cylinder which is $\dfrac{2}{3}$of TSA of the solid cylinder.
Now, we have to apply the formula to find the CSA of the solid cylinder which is mentioned below:
$ \Rightarrow $CSA$ = 2\pi rh......................(B)$
Now, we have to substitute all the values in the formula (A) and (B) and then we have to find the radius and height of the solid which can be obtained by the relation as mentioned in the question as CSA is $\dfrac{2}{3}$ of TSA.
Now, after obtaining height and radius of the solid cylinder we have to find the volume of the cylinder which can be determined with the help of the formula as mentioned below:
Formula used:
$ \Rightarrow $Volume of cylinder$ = \pi {r^2}h......................(C)$
Hence, with the help of the formula (C) above, we can determine the volume of the given solid cylinder.

Complete step-by-step solution:
Step 1: First of all we have to use the formula (A) as mentioned in the solution hint to determine the total surface area of the solid cylinder.
$ \Rightarrow 2\pi r(h + r) = 231c{m^2}................(1)$
Step 2: Now, we have to determine the CSA with the help of the formula (B) as mentioned in the solution hint. Hence,
$ \Rightarrow $CSA$ = 2\pi rh.............................(2)$
Step 3: Now, as mentioned in the question that the CSA is $\dfrac{2}{3}$of TSA so, we have to substitute all the values in the formula (A) and (B) as mentioned in the solution hint to obtain the radius and height of the solid cylinder. Hence,
$
   \Rightarrow 2\pi rh = \dfrac{2}{3} \times 231 \\
   \Rightarrow 2\pi rh = 154c{m^2}................(3)
 $
Step 4: Now, we have to substitute the expression (3) into expression (1) to obtain the radius of the solid cylinder as mentioned in the solution hint,
$
   \Rightarrow 2\pi rh + 2\pi {r^2} = 231 \\
   \Rightarrow 154 + 2\pi {r^2} = 231 \\
   \Rightarrow 2\pi {r^2} = 231 - 154 \\
   \Rightarrow 2\pi {r^2} = 77
 $
On applying cross-multiplication in the expression as obtained just above,
$ \Rightarrow {r^2} = \dfrac{{77}}{{2\pi }}$
On substituting the value of $\pi = \dfrac{{22}}{7}$in the expression as obtained just above,
$
   \Rightarrow {r^2} = \dfrac{{77 \times 7}}{{2 \times 22}} \\
   \Rightarrow {r^2} = \dfrac{{49}}{4} \\
   \Rightarrow r = \sqrt {\dfrac{{49}}{4}} \\
   \Rightarrow r = \dfrac{7}{2} \\
   \Rightarrow r = 3.5cm
 $
Step 5: Now, we have to substitute the value of radius (r) as obtained in the expression (3) as obtained in the solution step 3,
$ \Rightarrow 2 \times \dfrac{{22}}{7} \times \dfrac{7}{2} \times h = 154$
Applying cross-multiplication in the expression as obtained just above,
$
   \Rightarrow h = \dfrac{{154 \times 2 \times 7}}{{2 \times 22 \times 7}} \\
   \Rightarrow h = 7cm
 $
Step 6: Now, we have to obtain the volume of the solid cylinder with the help of the formula (C) as mentioned in the solution hint. Hence,
$
   \Rightarrow \dfrac{{22}}{7} \times \dfrac{7}{2} \times \dfrac{7}{2} \times 7 \\
   \Rightarrow 269.5c{m^3}
 $

The volume of the required cylinder is $269.5c{m^3}$

Note: To find the volume of the given solid cylinder it is necessary to find the radius and height of the given cone which is not mentioned in the question but we can determine the radius and height of the given cylinder with the relation between CSA and TSA of the solid cylinder.