Question

The ratio of The Total Surface area
of two Solid hemisphere is 16:9. find
the Ratio of their Volumes.​

Answer 1

Question:

The ratio of The Total Surface area

of two Solid hemisphere is 16:9. find

the Ratio of their Volumes.

Answer:

• The ratio of there volumes = 64:27

Given:

• The ratio of The Total Surface area
• of two Solid hemisphere is 16:9.

To find:

• Find the Ratio of their volumes?

Step by step explanation:

Let's use the proportion method and Let's do the solution.

$\sf Let \: the \: radius \: of \: 1st \: hemisphere \: be \: r_1 \\ \sf and \: radius \: of \: second \: hemisphere \: be \: r_2.$

Formula:

$\mathfrak{ \dfrac{Surface \: area \: of \: sphere \: r_1}{Surface \: area \: of \: Sphere \: r_2}= ratio}$

Solution:

By applying the formula , We get :

$: \implies \sf \dfrac{4\pi {r_1 }^{2} }{{4\pi r_2 }^{2}} = \dfrac{16}{9}$

Cut off 4π and 4π from up and down, Now we have:

$: \implies \sf \dfrac{ {r_1 }^{2} }{{ r_2 }^{2}} = \dfrac{16}{9}$

By simplifying, We are left with:

$: \implies \sf \dfrac{ {r _1 }^{2} }{{ r_2 }^{2}} = \dfrac{4}{3}$

This can also be written as :

$: \implies \sf {r_1 }^{2} : { r_2 }^{2} : : {4} : {3}$

According to the question,

$\sf Volume \: of \: hemisphere = \dfrac{ \dfrac{4}{3} \pi {r_1 }^{3} }{ \dfrac{4}{3}\pi { r_2 }^{3} } = \dfrac{4}{3}$

Cut off 4/3 , π from up and down, we get :

$: \implies \sf \dfrac{ {r_1 }^{3} }{ { r_2 }^{3} } = \dfrac{4}{3}$

Now, cube 4/3 , We get :

$: \implies \sf\dfrac{ {r_1 }^{3} }{ { r_2 }^{3} } = {\bigg (\dfrac{4}{3} \bigg)}^{3}$

By simplifying, we get :

$: \implies \sf\dfrac{ {r_1 }^{3} }{ { r_2 }^{3} } = \dfrac{64}{27}$

$: \implies \sf Ratio = 64:27$

Hence, The ratio of there volumes = 64:27.

Answer 2

Proper question:-

1. If the ratio of curved surface area of two solid spheres is 16:9. Find the ratio of their volumes.

To find,

• The volume of the solid spheres

Given that:-

• Surface area = 16:9
• Volume = ?

Required answer:-

• 64:27 is the volume required

Solution:-

The surface area of solid Sphere is 4πr²

According to the question,

$\rm\dfrac{4πr_1²}{4πr_2²}=\dfrac{16}{9}$

$\rm[{R_1 \: and \: R_2 \: are \: the \: radii \: of \: two \: solid \: sphere}]$

$\rm\dfrac{R_1}{R_2}=\dfrac{4}{3}$

The volume of solid Sphere is = 4/3πr³

→ The ratio of the volume of two solid spheres is,

$\rm\dfrac{4}{3}</strong><strong>π</strong><strong>r_1³;\dfrac{4</strong><strong>}</strong><strong>{3</strong><strong>π</strong><strong>r_2³$[tex]\rm = (\dfrac{R_1}{R_2})³=(\dfrac{4}{3})³=\dfrac{64}{27}[tex]

Answer → 64:27