Question

1) The diameter of a hemisphere is doubled of the diameter of a sphere. Find the ratio of volume of sphere and hemisphere.​

Answer 1

Given :

The diameter of a hemisphere is doubled of the diameter of a sphere.

To find :

Find the ratio of volume of sphere and hemisphere.​

Solution :

Let the diameter of sphere be d unit

∴ Diameter of hemisphere = 2d unit

So, radius of sphere = d/2 unit

And radius of hemisphere = 2d/2 = d unit

Now we know,

Volume of sphere = 4/3 πr³

Volume of hemisphere = 2/3 πr³

Now ratio of volume of sphere to volume of hemisphere :

⇒ Volume of sphere : Volume of hemisphere

⇒ 4/3 πr³ : 2/3 πr³

⇒ 4/3 π(d/2)³ : 2/3 π(d)³

⇒ { 4/3 π (d³/8) }/{2/3 πd³}

⇒ {πd³/6}/{2πd³/3}

⇒ (πd³/6) * (3/2πd³)

⇒ 1/(2 * 2)

⇒ 1/4

⇒ 1 : 4

Therefore,

Ratio of volume of sphere and hemisphere = 1 : 4

Answer 2

Given:-

• Diameter of a hemisphere is doubled of the diameter of shphere.

Find:-

• Ratio of volume of shpere and hemisphere.

Solution:-

Let, diameter of Sphere 'x' units

So, diameter of Hemisphere '2x' units

we, know that Radius is half of Diameter.

➜Radius of Shpere = x/2 units

➜Radius of Hemisphere = 2x/2 = x units

Now, using

$\huge{\underline{\boxed{\sf Volume \: of \: Sphere = \dfrac{4}{3}\pi {r}^{3}}}}$

$\pink{\sf where} \small{\begin{cases} \red{\sf r = \dfrac{x}{2}units} \\ \end{cases}}$

$\implies\sf Volume \: of \: Sphere = \dfrac{4}{3}\pi {r}^{3}$

$\implies\sf Volume \: of \: Sphere = \dfrac{4}{3} \times \pi \bigg({ \frac{x}{2} \bigg) }^{3}$

$\implies\sf Volume \: of \: Sphere = \dfrac{4}{3} \times \pi \times \dfrac{ {x}^{3} }{8}$

$\implies\sf Volume \: of \: Sphere = \dfrac{4 {x}^{3} }{24} \times \pi$

$\implies\sf Volume \: of \: Sphere = \dfrac{4 {x}^{3} \pi}{24} cube \: units$

Now, using

$\huge{\underline{\boxed{\sf Volume \: of \: Hemisphere = \dfrac{2}{3}\pi {r}^{3}}}}$

$\purple{\sf where} \small{\begin{cases} \orange{\sf r = xunits} \\ \end{cases}}$

$\implies\sf Volume \: of \: Hemisphere = \dfrac{2}{3}\pi {r}^{3}$

$\implies\sf Volume \: of \: Hemisphere = \dfrac{2}{3}\pi {(x)}^{3}$

$\implies\sf Volume \: of \: Hemisphere = \dfrac{2}{3} \times \pi \times {x}^{3}$

$\implies\sf Volume \: of \: Hemisphere = \dfrac{2 {x}^{3} }{3} \times \pi$

$\implies\sf Volume \: of \: Hemisphere = \dfrac{2 {x}^{3} \pi}{3}cube \: units$

Now, Ratio is

$\sf \dashrightarrow \dfrac{Volume\:of\:sphere}{Volume\:of\: Hemisphere}$

$\sf \dashrightarrow \dfrac{\dfrac{4 {x}^{3} \pi}{24}}{ \dfrac{2 {x}^{3} \pi}{3}}$

$\sf \dashrightarrow \dfrac{4 {x}^{3} \pi}{24} \times \dfrac{3}{2 {x}^{3} \pi}$

$\sf \dashrightarrow \dfrac{12{x}^{3} \pi}{48 {x}^{3}\pi}$

$\sf \dashrightarrow \dfrac{12}{48}$

$\sf \dashrightarrow \dfrac{1}{4}$

$\tiny{\therefore \underline{\textsf{Ratio of Surface Area of Sphere and Hemisphere is} \dfrac{1}{4} }}$