Question

if base radius of a right circular cylinder is halved keeping the height same find the ratio of the volume of the refused cylinder to that of the original cylinder

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Answer 1

☆Given Question : -

• If base radius of a right circular cylinder is halved keeping the height same, find the ratio of the volume of the reduced cylinder to that of the original cylinder.

☆Given :-

• The radius is reduced to half.
• Height remains the same.

☆To find :-

• Ratio of the volume of the reduced cylinder to that of the original cylinder.

☆ Formula Used :-

${{ \boxed{\large{\bold\green{Volume_{(Cylinder)}\: = \:\pi r^2 h }}}}}$

where,

• r = radius of cylinder
• h = height of cylinder

☆ Solution :-

☆ Case 1.

$\begin{gathered}\begin{gathered}\bf Let = \begin{cases} &\sf{radius \: of \: cylinder \: be \: r \: units} \\ &\sf{height \: of \: cylinder \: be \: h \: units} \end{cases}\end{gathered}\end{gathered}$

$\bf \:So, Volume \: \: of \: Cylinder ,V_1 = \pi \: {r}^{2} h \: ... \: (1)$

☆ Case 2.

$\begin{gathered}\begin{gathered}\bf Let = \begin{cases} &\sf{radius \: of \: cylinder \: be \: \dfrac{r}{2} \: units} \\ &\sf{height \: of \: cylinder \: be \: h \: units} \end{cases}\end{gathered}\end{gathered}$

$\bf \:So, Volume \: of \: Cylinder \: ,V_2 = \pi \: {(\dfrac{r}{2} )}^{2} h$

$\bf\implies \:V_2 = \pi \: \dfrac{ {r}^{2} }{4} h \: .... \: (2)$

☆ Now

$\bf \:Consider \: V_2 : V_1$

$\bf\implies \:\pi\dfrac{ {r}^{2} }{4} h : \pi \: {r}^{2} h$

$\bf\implies \:\dfrac{1}{4} : 1$

$\bf\implies \:1 : 4$

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Perimeter of rectangle = 2(length× breadth)

Diagonal of rectangle = √(length ²+breadth ²)

Area of square = side²

Perimeter of square = 4× side

Volume of cylinder = πr²h

T.S.A of cylinder = 2πrh + 2πr²

Volume of cone = ⅓ πr²h

C.S.A of cone = πrl

T.S.A of cone = πrl + πr²

Volume of cuboid = l × b × h

C.S.A of cuboid = 2(l + b)h

T.S.A of cuboid = 2(lb + bh + lh)

C.S.A of cube = 4a²

T.S.A of cube = 6a²

Volume of cube = a³

Volume of sphere = 4/3πr³

Surface area of sphere = 4πr²

Volume of hemisphere = ⅔ πr³

C.S.A of hemisphere = 2πr²

T.S.A of hemisphere = 3πr²