Introduction

Imagine you have 27 small spheres of the same size, and you decide to melt them down to create one larger sphere. How would you find the radius of this new sphere? In this article, we will walk through the process step by step using mathematical principles, ensuring you understand each part of the calculation.

Understanding the Problem

We start with 27 small spheres, each having a radius of ( r ). The first step is to calculate the volume of one small sphere.

Volume of One Small Sphere

The formula for the volume of a sphere is given by:

( V frac{4}{3} pi r^3 )

This is the volume of a single small sphere with radius ( r ).

Total Volume of 27 Small Spheres

Since we have 27 such spheres, the total volume of the small spheres is:

( V_{text{total}} 27 times frac{4}{3} pi r^3 36 pi r^3 )

This is the combined volume of the 27 small spheres.

Volume of the New Sphere

Let's denote the radius of the new sphere formed by melting the 27 small spheres as ( R ). The volume of this new sphere is given by:

( V_{text{new}} frac{4}{3} pi R^3 )

Setting the Volumes Equal

Since the volume of the 27 small spheres is the same as the volume of the new sphere, we set the two volumes equal:

( 36 pi r^3 frac{4}{3} pi R^3 )

By canceling out ( pi ) from both sides, we get:

( 36 r^3 frac{4}{3} R^3 )

Clearing the Fraction and Solving for ( R^3 )

Multiplying both sides by 3 to clear the fraction, we obtain:

( 108 r^3 4 R^3 )

Dividing both sides by 4, we then find:

( R^3 27 r^3 )

Finding ( R )

Finally, taking the cube root of both sides, we get:

( R sqrt[3]{27 r^3} 3r )

Conclusion

The radius of the new sphere, formed by melting 27 smaller spheres each of radius ( r ), is:

boxed{3r}

Key Takeaways:

The volume of the 27 small spheres equals the volume of the new sphere. The calculation involves using the formula for the volume of a sphere and setting it equal to find the radius of the new sphere. The final answer is ( R 3r ).