A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder. 

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A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder.

The height of the cylinder is approximately 2.744 cm.

To solve this problem, we need to use the principle of conservation of volume. The volume of the original sphere will be equal to the volume of the resulting cylinder.

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The volume of a sphere is given by the formula:

V_sphere = (4/3)πr^3

The volume of a cylinder is given by the formula:

V_cylinder = πr^2h

Given:

• Radius of the sphere, r_sphere = 4.2 cm
• Radius of the cylinder, r_cylinder = 6 cm

We first find the volume of the sphere and then equate it to the volume of the cylinder.

(4/3)π(4.2)^3 = π(6)^2h

Now, let’s solve for h, the height of the cylinder:

(4/3)π(4.2)^3 = π(6)^2h

(4/3)(4.2)^3 = 6^2h

(4/3)(4.2)^3 = 36h

h = ((4/3)(4.2)^3) / 36

h ≈ ((4/3)(74.088)) / 36

h ≈ (98.784) / 36

h ≈ 2.744

So, the height of the cylinder is approximately 2.744 cm.

Surface Areas And Volumes

Surface areas and volumes is a branch of mathematics that deals with calculating the total area and total volume of various geometric shapes such as cubes, spheres, cylinders, cones, and pyramids. Here’s a brief overview of some common formulas used in this topic:

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1. Cube:

   • Surface Area = 6a^2 (where ‘a’ is the length of a side)
   • Volume = a^3

2. Rectangular Prism:

   • Surface Area = 2(lw + lh + wh) (where ‘l’, ‘w’, and ‘h’ are the length, width, and height respectively)
   • Volume = lwh

3. Sphere:

   • Surface Area = 4πr^2 (where ‘r’ is the radius)
   • Volume = (4/3)πr^3

4. Cylinder:

   • Surface Area = 2πr^2 + 2πrh (where ‘r’ is the radius and ‘h’ is the height)
   • Volume = πr^2h

5. Cone:

   • Surface Area = πr^2 + πrl (where ‘r’ is the radius and ‘l’ is the slant height)
   • Volume = (1/3)πr^2h

6. Pyramid (for regular pyramid with a polygonal base):

   • Surface Area = Sum of areas of all faces
   • Volume = (1/3) × Base Area × Height

These are some basic formulas used to calculate surface areas and volumes of common geometric shapes.

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