Volume Conservation Principle

To solve this problem, we use the principle of volume conservation. When an object is moulded into another shape, the total volume of the material remains the same. Therefore, the total volume of all the small spheres must be equal to the volume of the resulting cylinder.

Number of Spheres (N) × Volume of one Sphere = Volume of the Cylinder

1. Given Dimensions and Radii

• Sphere:
  • Diameter = 3 cm
  • Radius ($r_s$) = 3 / 2 = 1.5 cm
• Cylinder:
  • Height (h) = 54 cm
  • Diameter = 4 cm
  • Radius ($r_c$) = 4 / 2 = 2 cm

2. Volume Formulas

• Volume of a Sphere = $\frac{4}{3}\pi r^3$
• Volume of a Cylinder = $\pi r^2 h$

3. Setting up and Solving the Equation

Let N be the number of spheres required.

$N \times (\frac{4}{3}\pi (r_s)^3) = \pi (r_c)^2 h$

We can cancel out $\pi$ from both sides:

$N \times (\frac{4}{3} \times (1.5)^3) = (2)^2 \times 54$

$N \times (\frac{4}{3} \times (\frac{3}{2})^3) = 4 \times 54$

$N \times (\frac{4}{3} \times \frac{27}{8}) = 216$

Simplify the fraction:

$N \times (\frac{108}{24}) = 216$

$N \times (\frac{9}{2}) = 216$

$N = 216 \times \frac{2}{9}$

$N = \frac{432}{9}$

$N = 48$

Conclusion

The number of solid spheres that could be moulded is 48.