Question

- A solid cylinder has a total surface area of 462 sq. cm. Its curved surface area is one-third of its total surface area. Find the volume of the cylinder.

•Don't spam
•QAN (Quality Answer Needed)
•Answer with full explanation
—————————————————​​

Answer 1

Given : A solid cylinder has a total surface area of 462 sq. cm and its Curved Surface Area is one-third of its Total Surface Area.

To Find : Volume of Cylinder ?

______________________

❍ Let's consider the Radius of Cylinder be r and Height of Cylinder be h.

$~$

$\underline{\frak{As~ we ~know~ that~:}}$

• $\boxed{\sf\pink{Total ~Surface ~Area_{(Cylinder)}~=~2~×~π~×~Radius(Height~+~Radius)~sq}}$

• $\boxed{\sf\pink{Curved ~Surface ~Area_{(Cylinder)}~=~2~×~π~×Radius~×~Height~sq}}$

$~$

Given that,

• It's Curved Surface Area is one-third of its Total Surface Area.

$~$

Then,

• $\boxed{\sf\pink{Total ~Surface ~Area_{(Cylinder)}~=~\dfrac{1}{3}(Total ~Surface ~Area_{(Cylinder)})}}$

$~$

Or,

• $\boxed{\sf\pink{2~×~π~×Radius~×~Height~=~\dfrac{1}{3}(2~×~π~×~Radius(Height~+~Radius))}}$

$~$

$\underline{\bf{Now ~By ~Substituting~ the ~Assumed ~Values~:}}$

$~$

${\sf{(2~×~π~×~r~×~h)~=~\dfrac{1}{3}(2~×~π~×~r(h~+~r))}}$

${\sf{3 ~× ~2 ~× ~π ~×~ r~ ×~ h~ = ~2~ × ~π ~× ~r(h + r)}}$

${\sf{6~ × ~π ~× ~r ~× ~h ~= ~2 ~×~ π~ × ~r(h ~+ ~r)}}$

${\sf{6~×~π~×~r~×~h~=~2~×~π~×~r~×~h~+~2~×~π~×~r^2}}$

${\sf{6 ~× ~π ~×~ r~ ×~ h ~- ~2 ~× ~π ~× ~r ~×~ h ~=~ 2~ ×~ π~ × ~r^2}}$

$:\implies{\sf{\cancel{4}~×~π~×~r~×~h~=~\cancel{2}~×~π~×~r^2}}$

$:\implies{\sf{2~×~\cancel{π}~×~r~×~h~=~\cancel{π}~×~r^2}}$

$:\implies{\sf{2~×~\cancel{r}~×~h~=~\cancel{r^2}~=~r}}$

$:\implies{\sf{2~×~h~=~r}}$

$:\implies{\underline{\boxed{\frak{\pink{r~=~2h}}}}}$★

_______________________________

Given that,

• A Solid Cylinder has a Total Surface Area of 462 sq. cm.

$~$

$\underline{\frak{As ~we~ know ~that~:}}$

• It's Curved Surface Area is one-third of its Total Surface Area.

• Curved Surface Area ${\sf{= \dfrac{1}{3}~×~462}}$

• Curved Surface Area ${\sf{=\dfrac{1}{\cancel{3}}~×~\cancel{462}}}$

• Curved Surface Area ${\bold{=~154~cm^2}}$

$~$

Or,

• ${\sf{2~×~π~×~r~×~h~=~\bf{154~cm^2}}}$

$~$

Here,

$~~~~~~~:\implies{\sf{2~×~h~=~r}}$

$~~~~~~~:\implies{\sf{2~×~\dfrac{22}{7}~×~2h~×~h~=~154~cm^2}}$

$~~~~~~~:\implies{\sf{2~×~\dfrac{22}{7}~×~2h^2~=~154~cm^2}}$

$~~~~~~~:\implies{\sf{h^2~=~\dfrac{154~×~7}{22~×~2~×~2}}}$

$~~~~~~~:\implies{\sf{h^2~=~\cancel\dfrac{1078}{88}}}$

$~~~~~~~:\implies{\sf{h^2~=~\dfrac{49}{4}}}$

$~~~~~~~:\implies{\sf{h~=~\sqrt{\dfrac{49}{4}}}}$

$~~~~~~~:\implies{\underline{\boxed{\frak{\pink{h~=~\dfrac{7}{2}~cm}}}}}$★

$~$

$\underline{\frak{As~ we ~know ~that ~:}}$

$~~~~~~~:\implies{\sf{2~×~h~=~r}}$

$~~~~~~~:\implies{\sf{\cancel{2}~×~\dfrac{7}{\cancel{2}}~=~r}}$

$~~~~~~~:\implies{\underline{\boxed{\frak{\pink{r~=~7~cm}}}}}$★

$~$

Therefore,

• Radius of Cylinder is ${\sf{:~h~=~\dfrac{7}{2}~cm}}$

• Height of Cylinder is ${\sf{:~h~=~7~cm}}$

_________________________

$~$

$\underline{\frak{As ~we~ know ~that~:}}$

• $\boxed{\sf\pink{Volume_{(Cylinder)}~=~2~×~π~Radius^2~×~Height~cu}}$

$~$

$\underline{\bf{Now ~By ~Substituting ~the ~Found ~Values ~:}}$

$~$

$\leadsto{\sf{Volume_{(Cylinder)}~=~\dfrac{22}{7}~×~7^2~×~\dfrac{7}{2}}}$

$\leadsto{\sf{Volume_{(Cylinder)}~=~\dfrac{22}{\cancel{7}}~×~\cancel{7}~×~7~×\dfrac{7}{2}}}$

$\leadsto{\underline{\boxed{\frak{\pink{Volume_{(Cylinder)}~=~539~cm^2}}}}}$★

$~$

Hence,

$\therefore\underline{\sf{Volume~of~Cylinder~is~\bf{539~cm^2}}}$

Answer 2

To find Length or Breadth when Area of a Rectangle is given

When we need to find length of a rectangle we need to divide area by breadth.

Length of a rectangle = Area ÷ breadth.

ℓ = A ÷ b.

Similarly, when we need to find breadth of a rectangle we need to divide area by length.

Breadth of a rectangle = Area ÷ length.

b = A ÷ ℓ

@BrainlyShinestar

This answer is above ryt.