Question

A metallic solid sphere of radius 10.5 cm is melted and recast into smaller solid cones, each of radius 3.5 cm and height 3 cm. Thus, how many such cones will be made?

Hint:

When a solid is recast into another shape, equate the volumes of the original and new shapes to find the required quantity.

Answer 1

The volume of the metallic sphere is given by the formula for the volume of a sphere: \[ V_{\text{sphere}} = \frac{4}{3} \pi r^3. \] Substitute \( r = 10.5 \, \text{cm} \): \[ V_{\text{sphere}} = \frac{4}{3} \pi (10.5)^3 = \frac{4}{3} \pi \times 1157.625 = 1543.5 \pi \, \text{cm}^3. \] The volume of one cone is given by the formula for the volume of a cone: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h. \] Substitute \( r = 3.5 \, \text{cm} \) and \( h = 3 \, \text{cm} \): \[ V_{\text{cone}} = \frac{1}{3} \pi (3.5)^2 \times 3 = \frac{1}{3} \pi \times 12.25 \times 3 = 12.25 \pi \, \text{cm}^3. \] Now, let \( N \) be the number of cones that can be made. Since the volume of the sphere is recast into the cones, we have: \[ N \times V_{\text{cone}} = V_{\text{sphere}}. \] Substitute the values of \( V_{\text{sphere}} \) and \( V_{\text{cone}} \): \[ N \times 12.25 \pi = 1543.5 \pi. \] Cancel \( \pi \) from both sides: \[ N \times 12.25 = 1543.5 \quad \Rightarrow \quad N = \frac{1543.5}{12.25} = 126. \]
Conclusion: The number of cones that can be made is 126.