Question

The curved surface area of a cylinder is 176 \[c{m^2}\] and its base area is 38.5 \[c{m^2}\]. Find the volume of the cylinder and justify your answer.

Answer 1

Hint: In this problem we are to find the volume of our given cylinder. We consider our radius of the cylinder as r and the height as h. After that we use the formulas of the surface area and base area of the cylinder to find r and h. Then we use the formula of the volume of the cylinder and reach our desired result.

Complete step-by-step answer:
Here we are given the surface area of a cylinder as, \[176{m^2}\] and base area is \[38.5c{m^2}\] .

Let us consider the given cylinder’s radius is r cm and the height is h cm.
The formula base area of cylinder, \[ = \pi {r^2} = 38.5c{m^2}\] (Given)
 \[ \Rightarrow \pi {r^2} = 38.5\]
Using, \[\pi = \dfrac{{22}}{7}\pi = \dfrac{{22}}{7}\] , we get,
 \[ \Rightarrow {r^2} = \dfrac{{38.5 \times 7}}{{22}}\]
On simplification we get,
 \[ \Rightarrow {r^2} = 12.25\]
On taking positive square root we get,
 \[ \Rightarrow r = \sqrt {12.25} \]
 \[ \Rightarrow r = 3.5cm\]
The formula of the curved surface area of cylinder \[ = 2\pi rh = 176c{m^2}\] (Given)
 \[\therefore 2\pi rh = 176\]
On substituting the value of r and \[\pi = \dfrac{{22}}{7}\pi = \dfrac{{22}}{7}\] , we get,
 \[ \Rightarrow 2 \times \dfrac{{22}}{7} \times 3.5 \times h = 176\]
On cross multiplication we get,
 \[ \Rightarrow h = \dfrac{{176 \times 7}}{{2 \times 22 \times 3.5}}\]
On simplification we get,
 \[ \Rightarrow h = 8cm\]
Therefore, we have, \[r = 3.5cm\] and \[h = 8cm\]
So, we have the volume of the cylinder,
 \[ = \pi {r^2}h\]
On substituting values we get,
 \[ = \dfrac{{22}}{7} \times {\left( {3.5} \right)^2} \times 8\]
On simplification we get,
 \[ = 308c{m^3}\]

Note: The volume which is needed in this problem is to be found in the unit \[c{m^3}\]. Point to be noted is that the unit should be taken care of as a proper unit. If there is a wrong unit given in the answer it would be considered as a wrong answer. Remember the volume of the cylinder.