Question

How many solid cylinders of radius 10 cm and height 6 cm can be made by melting a solid sphere of radius 30 cm?

Hint:

In problems involving melting and recasting, the key is to equate the volumes. Always write down the formulas for the volumes of the shapes involved and cancel out common terms like \(\pi\) before substituting the numbers to simplify the calculation.

Answer 1

Step 1: Understanding the Concept:
When an object is melted and recast into other objects, the total volume of the material remains constant. Therefore, the volume of the original sphere will be equal to the total volume of all the cylinders made from it.

Step 2: Key Formula or Approach:
1. Volume of a sphere = \( \frac{4}{3}\pi R^3 \), where R is the radius of the sphere.
2. Volume of a cylinder = \( \pi r^2 h \), where r is the radius and h is the height of the cylinder.
Let 'n' be the number of cylinders formed. Then, \[ \text{Volume of Sphere} = n \times \text{Volume of one Cylinder} \]

Step 3: Detailed Explanation:
Given for the sphere: Radius, R = 30 cm.
Given for the cylinder: Radius, r = 10 cm. Height, h = 6 cm.
Let 'n' be the number of cylinders that can be made.
Setting the volumes equal: \[ \frac{4}{3}\pi R^3 = n \times (\pi r^2 h) \] We can cancel \(\pi\) from both sides: \[ \frac{4}{3} R^3 = n \times r^2 h \] Substitute the given values: \[ \frac{4}{3} (30)^3 = n \times (10)^2 (6) \] \[ \frac{4}{3} \times (30 \times 30 \times 30) = n \times (100 \times 6) \] \[ 4 \times 10 \times 30 \times 30 = n \times 600 \] \[ 36000 = 600n \] Now, solve for n: \[ n = \frac{36000}{600} \] \[ n = \frac{360}{6} = 60 \]

Step 4: Final Answer:
The number of solid cylinders that can be made is 60.