Question

the volume of a cylinder is 2500π cm³ and its height is 49 cm. Find the total surface area and lateral surface area of the cylinder

Answer 1

Given :-

Volume of cylinder= 2500π cm³

Height of cylinder= 49cm

To Find :-

We have to find the TSA and CSA of cylinder

Solution :-

$\implies\sf\ TSA\ of\ cylinder= 2\pi r(h+r)\\ \\ \\ \sf\ CSA\ of\ cylinder= 2\pi r h$

Now ,

$\implies\sf\ Volume\ of\ cylinder= \pi r^2 h\\ \\ \\ :\implies\sf\ \cancel{\pi} r^2\times 49= 2500\cancel{\pi} \\ \\ \\ :\implies\sf\ r^2= \dfrac{2500}{49}\\ \\ \\ :\implies\sf\ r= \sqrt{\dfrac{2500}{49}}\\ \\ \\ :\implies\underline{\boxed{\sf\ r=\dfrac{50}{7}}} \ cm$

• Find the CSA of cylinder

$:\implies\sf\ CSA= 2\pi rh\\ \\ \\ :\implies\sf\ \ CSA= 2\times \dfrac{22}{7}\times \dfrac{50}{7}\times 49\\ \\ \\ :\implies\sf\ CSA= \dfrac{2\times 22\times 50\times \cancel{49}}{\cancel{7\times 7}}\\ \\ \\ :\implies\sf\ CSA= 44\times 50\\ \\ \\ \implies\underline{\boxed{\sf\ CSA=2200cm^2}}$

Now total surface area of cylinder

$:\implies\sf\ TSA= CSA+ 2\times Area\ of\ circle\\ \\ \\ :\implies\sf\ TSA= 2\pi r h+ 2(\pi r^2)\\ \\ \\ :\implies\sf\ TSA= 2200+ \bigg(2\times \dfrac{22}{7}\times \dfrac{50}{7}\times \dfrac{50}{7}\bigg)\\ \\ \\ :\implies\sf\ TSA= 2200+\bigg(\dfrac{44\times 2500}{49}\bigg)\\ \\ \\ :\implies\sf\ TSA=2200+\bigg(\cancel{\dfrac{110000}{49}}\bigg)\\ \\ \\ :\implies\sf\ TSA= 2200+2244.89\\ \\ \\ :\implies\underline{\boxed{\sf\ TSA= 4444.89cm^2}}$

Answer 2

Given :-

• The volume of a cylinder is 2500π cm³

• Height of cylinder is 49 cm

To Find :-

The total surface area and lateral surface area of the cylinder

Calculation :

Let the radius of the base and height of the cylinder be r cm and h cm respectively.

Then, h = 7cm ( given )

Now,

Volume = 2500π cm³

$⟹ \: πr^{2} h = 2500π \\ \\ ⟹ \: π \times {r}^{2} \times 49 = 2500π \\ \\ ⟹ {r}^{2} = \frac{2500}{49} \\ \\ ⟹ \: {r} = \frac{50}{7} \\ \\ ∴ \: lateral \: \: \: surface \: \: area = 2πrh \: {cm}^{2} \\ \\ = 2 \times \frac{22}{7} \times \frac{50}{7} \times 49 \\ \\ = 2200 \: {cm}^{2} \\ \\ total \: \: surface \: \: area = (2πrh + 2π {r}^{2} ) \: {cm}^{2} \\ \\ = 2πr(h + r) \: {cm}^{2} \\ \\ = 2200 + (2 \times \frac{22}{7} \times \frac{50}{7} \times \frac{50}{7} ) \\ \\ = 2200 + (\frac{11000}{49} ) \\ \\ = 2200 + 2244.89 \\ \\ = 4444.89 \: {cm}^{2}$