Question

Find the height and the total surface area of a right circular cylinder, if it’s curved surface area is $4400c{m^2}$ and circumference of the base is 110cm.

Answer 1

Hint: Here in this question we must know some formulas of right circular cylinder they are mentioned below: -
Volume of right circular cylinder=$\pi {r^2}h$
Lateral or curved surface area=$2\pi rh$
Total surface area=$2\pi {r^2} + 2\pi rh$

Complete step-by-step answer:
Draw a right circular cylinder with height ‘h’ and radius ‘r’ so that there is more clarity while solving this question.
                    

Now the curved surface area is given in question so we will use that value for making one equation and second equation we will make from base circumference.
Lateral or curved surface area=$2\pi rh$
\[ \Rightarrow 2\pi rh = 4400\]
$ \Rightarrow rh = \dfrac{{4400}}{{2\pi }}$ (Taking constant part one side)
$ \Rightarrow rh = \dfrac{{4400}}{{2 \times \dfrac{{22}}{7}}}$ (Putting value of$\pi = \dfrac{{22}}{7}$)
$ \Rightarrow rh = \dfrac{{4400}}{2} \times \dfrac{7}{{22}}$
(Writing in simplified manner and then cancelling numerator and denominator)
$\therefore rh = 700$ .................Equation (1)
Circumference of the base=110cm
From figure it is clear that base is circle and circumference of the circle is given by $2\pi r$
$ \Rightarrow 2\pi r = 110$
$ \Rightarrow r = \dfrac{{110}}{{2\pi }}$
\[ \Rightarrow r = \dfrac{{110}}{{2 \times \dfrac{{22}}{7}}}\] (Putting value of$\pi = \dfrac{{22}}{7}$)
\[ \Rightarrow r = \dfrac{{110 \times 7}}{{2 \times 22}}\]
\[ \Rightarrow r = \dfrac{{5 \times 7}}{2}\] (Writing in simplified manner and then cancelling numerator and denominator)
\[\therefore r = \dfrac{{35}}{2}cm\]
Putting value of radius into equation number1
$ \Rightarrow \dfrac{{35}}{2} \times h = 700$
$ \Rightarrow h = 700 \times \dfrac{2}{{35}}$ (Taking constant on one side)
$ \Rightarrow h = 20 \times 2$
$\therefore h = 40cm$
Now as we got both radius and height so we will apply total surface area formula: -
Total surface area=$2\pi {r^2} + 2\pi rh$
\[ \Rightarrow 2\pi {(\dfrac{{35}}{2})^2} + 2\pi \times \dfrac{{35}}{2} \times 40\]
\[ \Rightarrow 2\pi \times \dfrac{{35}}{2}(\dfrac{{35}}{2} + 40)\] (Taking \[2\pi \times \dfrac{{35}}{2}\]common from both terms )
\[ \Rightarrow \pi \times 35(\dfrac{{35 + 80}}{2})\]
\[ \Rightarrow \dfrac{{22}}{7} \times 35(\dfrac{{115}}{2})\] (Putting value of $\pi = \dfrac{{22}}{7}$)
\[ \Rightarrow 11 \times 5 \times 115\]

Therefore total surface area =\[6325c{m^2}\].

Note: In these types of questions of right circular cylinder students generally get confused between all the areas so it is very important to understand about all the areas.
Lateral or curved surface area= (Perimeter of the cross-section$ \times $Height)
Total surface area= Curved surface area+2$ \times $(Area of the cross-section)