Question

The volume of a cyilnder is V, curve area is A and radius is r. Then prove that 2V=Ar.​

Answer 1

$\huge\sf\blue{Given}$

✭ Volume of cylinder is V

✭ Curved Surface Area (CSA) is A

✭ Radius is r

━━━━━━━━━━━━━

$\huge\sf\gray{To \: Prove}$

◈ 2V = Ar

━━━━━━━━━━━━━

$\huge\sf\purple{Steps}$

We know that the volume of a cylinder is given by,

$\underline{\boxed{\sf V = \pi r^2 h}}$

So as per the question,

➝ $\sf V_{Cylinder}=\pi^2 rh$

➝ $\sf 2V = 2(\pi r^2h)$

➝ $\sf \red{2V = 2\pi r^2h}\:\:\: -eq(1)$

We also know that the the CSA of cylinder is given by,

$\underline{\boxed{\sf CSA_{Cylinder} = 2\pi rh}}$

Which can be written as,

➳ $\sf A = 2\pi rh$

$\bigg\lgroup\sf Multiply \ both \ sides \ by \ r\bigg\rgroup$

➳ $\sf A\times r = 2\pi r h \times r$

➳ $\sf \red{Ar = 2\pi r^2 h}\:\:\:-eq(2)$

Equating eq(1) & eq(2)

»» $\sf 2\pi r^2 h = 2\pi r^2 h$

»» $\sf 2V = Ar$

$\sf Hence \ Proved!!!$

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Answer 2

❒ How to solve ?

Here, We will take LHS as volume of cylinder, then we will multiply both sides by 2, After that we will take RHS as curved surface area, then we will multiply it by r (radius) on both the sides. Thus on solving, we will find that LHS = RHS i.e.,

2V = Ar ☑

♕ ${ \gray{ \underline{ \underline{ \rm{\huge{ \red{S} \pink{o} \purple{lu} \orange{ti} \green{on}}}}}}}$ ♕

${ \blue{ \underline{ \underline{ \rm{ \purple{ Given : }}}}}}$

◑ Volume of a cylinder = V

◑ Curved Surface Area (CSA) = A

◑ Radius = r

$\blue{ \underline{ \underline{ \rm{ \purple{To \: Prove : }}}}}$

❍ 2V = Ar

$\blue{ \underline{ \underline{ \rm{ \purple{Let's \: Prove \: it : }}}}}$

$\blue{ \sf{ \underline{\large{LHS}}}}$

Volume of cylinder = Area of cross-section × height

${ \blue{ \rm{ V = \pi {r}^{2} \times h}}}$

$\green{ \underline{ \boxed{ \blue{ \rm{ V = \pi {r}^{2} h}}}}}$

（Multiple 2 on both sides）

$\blue{ \rm{2 \times V = 2 \times \pi {r}^{2} h}}$

$\blue{ \rm{2V = 2\pi {r}^{2}h}}$ ........①

$\blue{ \sf{\large{ \underline{RHS}}}}$

Curved surface area = Perimeter of cross-section × height

$\blue{ \rm{A = 2\pi r \times h}}$

$\green{ \underline{ \boxed{ \blue{ \rm{A = 2\pi rh}}}}}$

（Multiply r on both sides）

$\blue{ \rm{A \times r = 2\pi rh \times r}}$

$\blue{ \rm{Ar = 2\pi {r}^{2}h}}$ ........②

Thus, from equation ① and ② we have,

${ \purple{ \rm{2\pi {r}^{2}h = 2\pi {r}^{2} h}}}$

$</p><p>LHS = RHS</p><p> \green{ \underline{ \overline{ \boxed{ \blue{ \rm{ \therefore{2V = Ar}}}}}}}$

${ \gray{ \tt{Hence \: Proved !!}}}$

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