Question

A hemispherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into cylindrical shaped small bottles each of diameter 3 cm and height 4 cm. How many bottles are necessary to empty the bowl?

Answer 1

The number of bottles necessary to empty the bowl is 54 bottles

Step-by-step explanation:

Given as :

The internal radius of hemispherical bowl full of liquid = r = 9 cm

The liquid is to be filled in cylinder shaped bottles

The diameter of cylindrical bottles = D = 3 cm

So, radius of cylindrical bottle = r = $\dfrac{D}{2}$   =  $\dfrac{3}{2}$ = 1.5 cm

The height of cylindrical bottle = h = 4 cm

Let The number of bottles necessary to empty the bowl = n bottles

According to question

Volume of hollow hemispherical shell = V = $\dfrac{2}{3}$ × π × r³  

Or,                                                      V = $\dfrac{2}{3}$ × π ×  9³

Or,                                                      V = $\dfrac{2}{3}$ × 3.14 × 729

Or,                                                      V = 1526.04  cm³

Again

From hemispherical bowl liquid is filled into a cylinder bottle

Volume of cylinder = π × R² × h

Or,                         v = 3.14 × 1.5² × 4

Or,                         v = 28.26 cm³

As From hemispherical bowl liquid is filled into a cylinder bottle

So, Number of bottle = $\dfrac{volume of hemispherical bowl}{volume of cylindrical bottle}$

Or, n = $\dfrac{V}{v}$

Or,   n = $\dfrac{1526.04}{28.26}$

i.e    n = 54

So, The number of bottles necessary to empty the bowl = n = 54 bottles

Hence, The number of bottles necessary to empty the bowl is 54 bottles . Answer

Answer 2

54 cylindrical bottles are necessary to empty the hemispherical bowl.

• Given,

Internal radius of the hemispheric bowl (r) = 9 cm

Diameter of the cylindrical bottle = 3 cm

=> Radius of the bottle = D / 2 = 3 cm / 2

Height of the cylindrical bottle = 4 cm

Number of cylindrical bottles required = ?

• Volume of water contained in the hemispherical bowl = Volume of the bowl

• Volume of a hemisphere is given by (2 / 3).π.(radius)³.

Volume of the hemispherical bowl = (2 / 3).π.(9 cm)³

= (2 × π × 9 × 9 × 9) cm³ / 3

= (2 × π × 3 × 9 × 9) cm³

= 486π cm³

• Volume of a cylinder is given by π.(radius)².height

Volume of each cylindrical bottle = π.(3 cm / 2)².4 cm

= (π × 3 cm × 3 cm × 4 cm) / (2 × 2)

= π × 3 cm × 3 cm × 1 cm

= 9π cm³

• Let the number of cylindrical bottles required be x.

• According to the question,

486π cm³ = x × 9π cm³

=> x = 486π cm³ / 9π cm³

=> x = 54

Therefore, 54 cylindrical bottles are required.