Question

(iv) If the ratio of curved surface areas of two solid spheres is 16:9, the ratio of their volumes is
(a) 64:27 (b) 4:3 (c) 27:64 (d) 3:4​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that radius of first sphere be 'x' units and radius of second sphere be 'y' units.

Let assume that

$\rm :\longmapsto\:S_1 \: be \: surface \: area \: of \: first \: sphere$

and

$\rm :\longmapsto\:S_2 \: be \: surface \: area \: of \: second \: sphere$

Let further assume that,

$\rm :\longmapsto\:V_1 \: be \: volume\: of \: first \: sphere$

and

$\rm :\longmapsto\:V_2 \: be \: volume\: of \: second \: sphere$

We know,

Surface area and Volume of sphere of radius r is given by

$\red{\boxed{ \rm{ \: Surface \: area_{(sphere)} \: = \: 4 \: \pi \: {r}^{2} \: \: }}}$

and

$\red{\boxed{ \rm{ \: Volume_{(sphere)} \: = \: \frac{4}{3} \: \pi \: {r}^{3} \: \: }}}$

According to statement,

$\purple{\rm :\longmapsto\: \bf{ \: \dfrac{S_1}{S_2} = \dfrac{16}{9} }}$

$\purple{\rm :\longmapsto\: \bf{ \: \dfrac{4\pi {x}^{2} }{4\pi {y}^{2} } = \dfrac{16}{9} }}$

$\purple{\rm :\longmapsto\: \bf{ \: \dfrac{ {x}^{2} }{{y}^{2} } = \dfrac{16}{9} }}$

$\purple{\rm :\longmapsto\: \bf{ \: \dfrac{ {x}}{{y}} = \dfrac{4}{3} }}$

On cubing both sides, we get

$\purple{\rm :\longmapsto\: \bf{ \: \dfrac{ { {x}^{3} }}{{ {y}^{3} }} = \dfrac{64}{27} }}$

can be rewritten as

$\purple{\rm :\longmapsto\: \bf{ \: \dfrac{ \dfrac{4}{3} \pi \: { {x}^{3} }}{\dfrac{4}{3} \pi \: { {y}^{3} }} = \dfrac{64}{27} }}$

$\purple{\rm :\longmapsto\: \bf{ \: \dfrac{V_1 }{V_2} = \dfrac{64}{27} }}$

$\purple{ \bf\implies\bf{ \: V_1 : V_2 \: = \: 64 : 27 \: \: }}$

• Hence, Option ( a ) is correct.

More information :

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

Answer 2

Answer:

B) 16:9

Step-by-step explanation:

Volume of sphere = \frac{4}{3\\}  * π * r^3

Ratio of volume of 2 spheres =   4/3 * pi * r^3 / 4/3 * pi * R^3  

= r^3 / R^3

 The ratio is given 64:27  

r/R = 4/3  

Curved surface area of sphere = 4πr^2  

Ratio of their CSA = r^2 / R^2  

= 16:9

thanks..