Question

the no of solid sphere each of diameter 6cm that could be moulded to from a solid metes cylinder of height 45cm and diameter 4cm​

Answer 1

Given :

• The no of solid sphere each of diameter 6cm that could be moulded to from a solid metes cylinder of height 45cm and diameter 4cm.

To Find :

• Number of solid sphere = ?

Solution :

• Diameter of the solid sphere = 6 cm
• Height of solid metal cylinder = 45 cm
• Diameter of solid metal cylinder = 4 cm.

❍ Finding the radius of solid sphere :

→ Radius of solid sphere = Diameter of solid sphere ÷ 2

→ Radius of solid sphere = 6 ÷ 2

→ Radius of solid sphere = 3 cm

• Hence, the radius of the solid sphere is 3 cm.

❍ Finding the radius of solid metal cylinder :

➻ Radius of solid metal cylinder = Diameter of solid metal cylinder ÷ 2

➻ Radius of solid metal cylinder = 4 ÷ 2

➻ Radius of solid metal cylinder = 2 cm

• Hence, the radius of the solid metal cylinder is 3 cm.

❍ Finding volume of the solid metal cylinder :

➥ Volume of cylinder = πr²h

➥ Volume of cylinder = π × (2)² × 45

➥ Volume of cylinder = π × 4 × 45

➥ Volume of cylinder = 180π cm³

• Hence,the volume of the solid metal cylinder is 180π cm³.

❍ Finding volume of the solid sphere :

➦ Volume of sphere = 4/3πr³

➦ Volume of sphere = 4/3 × π × (3)³

➦ Volume of sphere = 4/3 × π × 27

➦ Volume of sphere = 4 × π × 9

➦ Volume of sphere = 36π cm³

• Hence,the volume of the solid sphere is 36π cm³.

❍ Now, Let's find the total number of solid sphere:

Let the total number of solid sphere be 'x'

★ According to Question now :

⋙ Volume of cylinder = x * Volume of sphere

⋙ x = Volume of cylinder ÷ Volume of sphere

⋙ x = 180π ÷ 36π

Canceling π from numerator as well as from denomintaor :

⋙ x = 180 ÷ 36

⋙ x = 5

• Hence, the total number of solid sphere is 5.

Answer 2

$\large{\boxed{\boxed{ \sf Let's \: Understand \: Question \: F1^{st}}}}$

Here, we have given diameter of solid sphere which can be moulded to form a solid cylinder of given height and diameter and have to find the no. of sphere needed to make the cylinder of given diameter.

$\large{\boxed{\boxed{ \sf How \: To \: Do \: It?}}}$

Here, f1st we simply find the radius of the sphere and cylinder which can be made later then applying the formula for volume for sphere and cylinder we can find the volume both fig. then dividing volume of Cylinder by volume of sphere we can obtain the no. of solid sphere needed to make the given cylinder which is our required answer.

Let's Do It ⚡

$\huge{\underline{\boxed{ \sf AnSwer}}}$

_____________________________

Given:-

• Diameter of sphere = 6cm
• Height of cylinder = 45cm
• Diameter of Cylinder = 4cm

Find:-

• No. of solid sphere needed to melt to form a solid metes cylinder.

Solution:-

$\underline{\underline{\pink{\maltese\textsf{\: For Sphere \: }}}}$

⊍ Diameter of Sphere, D = 6cm

⊍ Radius, R = Diameter/2 = 6/2 = 3cm

⊍ Radius, R = 3cm

Now, using

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

$\sf where \small{\begin{cases} \sf R = 3cm \\ \end{cases}}$

$\blue\bigstar$ Substituting this values:

$:\implies\sf Volume_{Sphere} = \dfrac{4}{3} \pi R^{3}$

$:\implies\sf Volume_{Sphere} = \dfrac{4}{3} \pi \times (3)^{3}$

$:\implies\sf Volume_{Sphere} = \dfrac{4}{3} \pi \times 27$

$\qquad$ _______________________

$\underline{\underline{\purple{\maltese\textsf{\: For Cylinder \: }}}}$

⊎ Diameter of cylinder, d = 4cm

⊎ Radius, r = d/2 = 4/2 = 2cm

⊎ Radius, r = 2cm

Now, using

$\huge{\underline{\boxed{ \sf Volume \: of \: Cylinder = \pi {r}^{2}h}}}$

$\sf where \small{\begin{cases} \sf r= 2cm \\ \sf h = 45cm\end{cases}}$

$\orange\bigstar$ Substituting this value:

$:\implies\sf Volume_{Cylinder} = \pi {r}^{2}h$

$:\implies\sf Volume_{Cylinder} = \pi {(2)}^{2} \times (45)$

$:\implies\sf Volume_{Cylinder} = \pi \times 4 \times 45$

$\qquad$ _______________________

$\underline{\underline{\red{\maltese\textsf{\: No. of sphere \: }}}}$

Let, no. of sphere be 'n'

Then,

$\sf \dashrightarrow Volume \: of \: Cylinder = n\times Volume \: of \: sphere$

$\sf \dashrightarrow \dfrac{Volume \: of \: Cylinder}{Volume \: of \: sphere }= n$

$\sf \dashrightarrow \dfrac{\pi \times 4 \times 45}{\dfrac{4}{3} \pi \times 27}= n$

$\sf \dashrightarrow \dfrac{\pi \times 180}{\dfrac{108}{3} \pi}= n$

$\sf \dashrightarrow \dfrac{ \not\pi \times 180}{\dfrac{108}{3} \not\pi}= n$

$\sf \dashrightarrow \dfrac{180}{\dfrac{108}{3}}= n$

$\sf \dashrightarrow 180 \times \dfrac{3}{108} = n$

$\sf \dashrightarrow \dfrac{540}{108} = n$

$\sf \dashrightarrow 5= n$

$\underline{\boxed{ \sf \therefore No.\: Sphere\:needed\:to\:make\:the\: cylinder\:is\:5}}$