Understanding the Sphere Melting Problem

This problem involves the concept of volume conservation. When smaller objects are melted and reformed into a larger object, the total volume of the material remains the same. In this case, 216 identical solid spheres are melted to form one larger sphere. We need to find the diameter of this larger sphere.

Key Information

• Number of small spheres ($n$): 216
• Radius of each small sphere ($r$): 7 cm
• Shape: Solid spheres
• Goal: Find the diameter of the large sphere formed.

Volume Formula for a Sphere

The volume ($V$) of a sphere with radius ($R$) is given by the formula: $V = \frac{4}{3} \pi R^3$

Step-by-Step Solution

1. Volume of One Small Sphere

First, calculate the volume of a single small sphere using the given radius ($r = 7$ cm). $V_{small} = \frac{4}{3} \pi r^3$ $V_{small} = \frac{4}{3} \pi (7 \text{ cm})^3$ $V_{small} = \frac{4}{3} \pi (343) \text{ cm}^3$

2. Total Volume of Small Spheres

Next, find the total volume of all 216 small spheres by multiplying the volume of one small sphere by 216. $V_{total} = n \times V_{small}$ $V_{total} = 216 \times \left( \frac{4}{3} \pi (343) \text{ cm}^3 \right)$

3. Volume of the Big Sphere

Let the radius of the big sphere be $R_{big}$. Its volume ($V_{big}$) is given by: $V_{big} = \frac{4}{3} \pi R_{big}^3$

4. Equating Volumes

According to the principle of volume conservation, the total volume of the small spheres equals the volume of the big sphere: $V_{total} = V_{big}$ $216 \times \left( \frac{4}{3} \pi (7)^3 \right) = \frac{4}{3} \pi R_{big}^3$

5. Solving for the Radius of the Big Sphere ($R_{big}$)

We can simplify the equation by cancelling out the common terms ($\frac{4}{3} \pi$) from both sides: $216 \times (7)^3 = R_{big}^3$

Recognize that $216$ is a perfect cube: $216 = 6^3$. $6^3 \times 7^3 = R_{big}^3$

Using the property of exponents $(a \times b)^m = a^m \times b^m$, we can rewrite the left side: $(6 \times 7)^3 = R_{big}^3$ $42^3 = R_{big}^3$

Taking the cube root of both sides gives us the radius of the big sphere: $R_{big} = 42$ cm.

6. Calculating the Diameter of the Big Sphere

The diameter ($D_{big}$) is twice the radius. $D_{big} = 2 \times R_{big}$ $D_{big} = 2 \times 42 \text{ cm}$ $D_{big} = 84 \text{ cm}$

Conclusion

Therefore, the diameter of the big sphere formed by melting 216 identical solid spheres of radius 7 cm is 84 cm.