Question

Metal spheres each of radius 2 cm , are packed into a rectangular box of internal dimensions 16 cm.8 cm.8 cm. when 16 spheres are packed the sphere is filled with preservative liquid . find the volume of the liquid. (take pie=3.14)

please answer step by step please and no extra comment please....​

Answer 1

☆ To Find :

The volume of the liquid kept in the Rectangular Box.

☆ We Know :

☞ Volume of a Sphere :

$\purple{\large\sf{\underline{\boxed{V = \dfrac{4}{3}\pi r^{3}}}}}$

Where r is the radius of the Sphere.

☞ Volume of a Cuboid :

$\purple{\large\sf{\underline{\boxed{V = l \times b \times h}}}}$

Where ,

$\bullet$ l = length of the Cuboid.

$\bullet$ b = breadth of the Cuboid.

$\bullet$ h = height of the Cuboid.

☆ Solution :

☞ Given :

• Radius of the Sphere = 2 cm

• Length of the Cuboid = 16 cm

• Breadth of the Cuboid = 8 cm

• Height of the Cuboid = 8 cm

☞ Concept :

According to the question , we have to find the volume of the liquid contained in the Rectangular Box.

The volume of the liquid will be Equal to the difference of the volume of the Cuboid and volume of the 16 Spheres.

$\therefore$Volume of Liquid = Volume of 16 sphere - Volume of the Cuboid.

☞ Calculation :

Volume of 16 Spheres :

• r = 2 cm

Using the formula and substituting the values in it , we get :

$\purple{\sf{V = 16 \times \dfrac{4}{3}\pi r^{3}}} \\ \\ \\ \implies \sf{V = 16 \times \dfrac{4}{3} \times 3.14 \times 2^{3}} \\ \\ \\ \implies \sf{V = 16 \times \dfrac{4}{3} \times 3.14 \times 8} \\ \\ \\ \implies \sf{V = 16 \times \dfrac{4}{3} \times 25.12} \\ \\ \\ \implies \sf{V = 16 \times \dfrac{100.48}{3}} \\ \\ \\ \implies \sf{V = \dfrac{1607.68}{3}} \\ \\ \\ \implies \sf{V = 535.89 cm^{3}} \\ \\ \\ \therefore \purple{\sf{V = 535.89 cm^{3}}}$

Hence ,the Volume of the Sphere is 535.89 cm³

Volume of the Rectangular Box or Cuboid :

• Length of the Cuboid = 16 cm

• Breadth of the Cuboid = 8 cm

• Height of the Cuboid = 8 cm

Using the formula and substituting the values in it , we get :

$\purple{\sf{V = l \times b \times h}} \\ \\ \\ \implies \sf{V = 16 \times 8 \times 8} \\ \\ \\ \implies \sf{V = 16 \times 64} \\ \\ \\ \implies \sf{V = 1024 cm^{3}} \\ \\ \\ \therefore \purple{\sf{V = 1024 cm^{3}}}$

Hence ,the Volume of the Cuboid is 1024 cm³.

Volume of the Liquid :

We Know that , the Volume of the Liquid will be Equal to the difference of the Volume of the Cuboid and the volume of the 16 Spheres.

$\purple{\sf{V_{(liquid)}} = V_{(Cuboid)} - V_{(Sphere)}} \\ \\ \\ \implies \sf{V_{l} = (1024 - 535.89)} \\ \\ \\ \implies \sf{V_{l} = 488.11 cm^{3}} \\ \\ \\ \therefore \purple{\sf{V_{l} = 488.11 cm^{3}}}$

Hence , the volume of the Liquid is 488.11 cm³.

☆ Additional information :

• Volume of a Cylinder = πr²h

• Volume of a Cube = a³

• TSA of a hemisphere = 3πr³